Definitions
Standard normal distribution
The simplest case of a normal distribution is known as the ''standard normal distribution'' or ''unit normal distribution''. This is a special case when and , and it is described by this probability density function (or density): : The variable has a mean of 0 and a variance and standard deviation of 1. The density has its peak at and inflection points at and . Although the density above is most commonly known as the ''standard normal,'' a few authors have used that term to describe other versions of the normal distribution.General normal distribution
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor (the standard deviation) and then translated by (the mean value): : The probability density must be scaled by so that the integral is still 1. If is aNotation
The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter ( phi). The alternative form of the Greek letter phi, , is also used quite often. The normal distribution is often referred to as or . Thus when a random variable is normally distributed with mean and standard deviation , one may write :Alternative parameterizations
Some authors advocate using the precision as the parameter defining the width of the distribution, instead of the deviation or the variance . The precision is normally defined as the reciprocal of the variance, . The formula for the distribution then becomes : This choice is claimed to have advantages in numerical computations when is very close to zero, and simplifies formulas in some contexts, such as in theCumulative distribution functions
TheStandard deviation and coverage
Quantile function
TheProperties
The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance. Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.Geary RC(1936) The distribution of the "Student's" ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184 The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or theSymmetries and derivatives
The normal distribution with densityMoments
The plain and absolute moments of a variableFourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Fourier transform and characteristic function
TheMoment and cumulant generating functions
TheStein operator and class
Within Stein's method the Stein operator and class of a random variableZero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*Zero-variance limit
In the limit (mathematics), limit whenMaximum entropy
Of all probability distributions over the reals with a specified meanOther properties
Related distributions
Central limit theorem
Operations and functions of normal variables
Operations on a single normal variable
If= Operations on two independent normal variables
= * If= Operations on two independent standard normal variables
= IfOperations on multiple independent normal variables
* Any linear combination of independent normal deviates is a normal deviate. * IfOperations on multiple correlated normal variables
* AOperations on the density function
TheInfinite divisibility and Cramér's theorem
For any positive integerBernstein's theorem
Bernstein's theorem states that ifExtensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists. * The multivariate normal distribution describes the Gaussian law in the ''k''-dimensionalStatistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want toSample mean
EstimatorSample variance
The estimatorConfidence intervals
ByNormality tests
Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically theBayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: * Either the mean, or the variance, or neither, may be considered a fixed quantity. * When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified. * Both univariate andSum of two quadratics
= Scalar form
= The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious. := Vector form
= A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B areSum of differences from the mean
Another useful formula is as follows:With known variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith known mean
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsWith unknown mean and unknown variance
For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' followsOccurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories: # Exactly normal distributions; # Approximately normal laws, for example when such approximation is justified by the central limit theorem; and # Distributions modeled as normal – the normal distribution being the distribution with Principle of maximum entropy, maximum entropy for a given mean and variance. # Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.Exact normality
Approximate normality
''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects. * In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where Infinite divisibility, infinitely divisible and Indecomposable distribution, decomposable distributions are involved, such as ** binomial distribution, Binomial random variables, associated with binary response variables; ** Poisson distribution, Poisson random variables, associated with rare events; * Thermal radiation has a Bose–Einstein statistics, Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.Assumed normality
Methodological problems and peer review
John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiability, falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.Computational methods
Generating values from normal distribution
Numerical approximations for the normal CDF and normal quantile function
The standard normal cumulative distribution function, CDF is widely used in scientific and statistical computing. The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and Gauss's continued fraction#Of Kummer's confluent hypergeometric function, continued fractions. Different approximations are used depending on the desired level of accuracy. * give the approximation for Φ(''x'') for ''x > 0'' with the absolute error (algorithHistory
Development
Some authors attribute the credit for the discovery of the normal distribution to Abraham de Moivre, de Moivre, who in 1738 published in the second edition of his "''The Doctrine of Chances''" the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude ofNaming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, Gaussian law. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Charles Sanders Peirce, Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Karl Pearson, Pearson popularized the term ''normal'' as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Ronald Fisher, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:See also
* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range * Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means; * Bhattacharyya distance – method used to separate mixtures of normal distributions * Erdős–Kac theorem – on the occurrence of the normal distribution in number theory * Full width at half maximum * Gaussian blur –Notes
References
Citations
Sources
* * In particular, the entries foExternal links
*