In
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s that arise when estimating the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
of a
normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using ...
in situations where the
sample size
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
is small and the population's
standard deviation is unknown. It was developed by English statistician
William Sealy Gosset
William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small sam ...
under the pseudonym "Student".
The ''t''-distribution plays a role in a number of widely used statistical analyses, including
Student's ''t''-test for assessing the
statistical significance
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
of the difference between two sample means, the construction of
confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
s for the difference between two population means, and in linear
regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
. Student's ''t''-distribution also arises in the
Bayesian analysis
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
of data from a normal family.
If we take a sample of
observations from a normal distribution, then the ''t''-distribution with
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
can be defined as the distribution of the location of the sample mean relative to the true mean, divided by the sample standard deviation, after multiplying by the standardizing term
. In this way, the ''t''-distribution can be used to construct a
confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
for the true mean.
The ''t''-distribution is symmetric and bell-shaped, like the normal distribution. However, the ''t''-distribution has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's ''t''-distribution is a special case of the
generalized hyperbolic distribution.
History and etymology

In statistics, the ''t''-distribution was first derived as a
posterior distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
in 1876 by
Helmert
Friedrich Robert Helmert (31 July 1843 – 15 June 1917) was a German geodesist and statistician with important contributions to the theory of errors.
Career
Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg an ...
and
Lüroth.
The ''t''-distribution also appeared in a more general form as
Pearson Type IV distribution in
Karl Pearson
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
's 1895 paper.
In the English-language literature, the distribution takes its name from William Sealy Gosset's 1908 paper in ''
Biometrika
''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for thBiometrika Trust The editor-in-chief is Paul Fearnhead ( Lancaster University). The principal focus of this journal is theoretical statistics. It was ...
'' under the pseudonym "Student". One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Another version is that Guinness did not want their competitors to know that they were using the ''t''-test to determine the quality of raw material.
Gosset worked at the
Guinness Brewery
St. James's Gate Brewery is a brewery founded in 1759 in Dublin, Ireland, by Arthur Guinness. The company is now a part of Diageo, a company formed from the merger of Guinness and Grand Metropolitan in 1997. The main product of the brewery is G ...
in
Dublin, Ireland
Dublin (; , or ) is the capital and largest city of Ireland. On a bay at the mouth of the River Liffey, it is in the province of Leinster, bordered on the south by the Dublin Mountains, a part of the Wicklow Mountains range. At the 2016 c ...
, and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became well known through the work of
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
, who called the distribution "Student's distribution" and represented the test value with the letter ''t''.
How Student's distribution arises from sampling
Let
be independently and identically drawn from the distribution
, i.e. this is a sample of size
from a normally distributed population with expected mean value
and variance
.
Let
:
be the sample mean and let
:
be the (
Bessel-corrected) sample variance. Then the random variable
:
has a standard normal distribution (i.e. normal with expected mean 0 and variance 1), and the random variable
:
''i.e'' where
has been substituted for
, has a Student's ''t''-distribution with
degrees of freedom. Since
has replaced
the only unobservable quantity in this expression is
so this can be used to derive confidence intervals for
The numerator and the denominator in the preceding expression are
statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of ...
random variables despite being based on the same sample
. This can be seen by observing that
and recalling that
and
are both linear combinations of the same set of i.i.d. normally distributed random variables.
Definition
Probability density function
Student's ''t''-distribution has the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(PDF) given by
:
where
is the number of ''
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
'' and
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
. This may also be written as
:
where B is the
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^ ...
. In particular for integer valued degrees of freedom
we have:
For
even,
:
For
odd,
:
The probability density function is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the ''t''-distribution approaches the normal distribution with mean 0 and variance 1. For this reason
is also known as the normality parameter.
The following images show the density of the ''t''-distribution for increasing values of
. The normal distribution is shown as a blue line for comparison. Note that the ''t''-distribution (red line) becomes closer to the normal distribution as
increases.
Cumulative distribution function
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
(CDF) can be written in terms of ''I'', the regularized
incomplete beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t ...
. For ''t'' > 0,
[
:
where
:
Other values would be obtained by symmetry. An alternative formula, valid for , is][
:
where 2''F''1 is a particular case of the hypergeometric function.
For information on its inverse cumulative distribution function, see .
]
Special cases
Certain values of give a simple form for Student's t-distribution.
How the ''t''-distribution arises
Sampling distribution
Let be the numbers observed in a sample from a continuously distributed population with expected value . The sample mean and sample variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
are given by:
:
The resulting ''t-value'' is
:
The ''t''-distribution with degrees of freedom is the sampling distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were se ...
of the ''t''-value when the samples consist of independent identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
observations from a normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
population. Thus for inference purposes ''t'' is a useful "pivotal quantity
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity nee ...
" in the case when the mean and variance are unknown population parameters, in the sense that the ''t''-value has then a probability distribution that depends on neither nor .
Bayesian inference
In Bayesian statistics, a (scaled, shifted) ''t''-distribution arises as the marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
of the unknown mean of a normal distribution, when the dependence on an unknown variance has been marginalized out:
:
where stands for the data , and represents any other information that may have been used to create the model. The distribution is thus the compounding
In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
of the conditional distribution of given the data and with the marginal distribution of given the data.
With data points, if uninformative, or flat, the location prior can be taken for ''μ'', and the scale prior can be taken for ''σ''2, then Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...
gives
:
a normal distribution and a scaled inverse chi-squared distribution respectively, where and
:
The marginalization integral thus becomes
:
This can be evaluated by substituting , where , giving
:
so
:
But the ''z'' integral is now a standard Gamma integral, which evaluates to a constant, leaving
:
This is a form of the ''t''-distribution with an explicit scaling and shifting that will be explored in more detail in a further section below. It can be related to the standardized ''t''-distribution by the substitution
:
The derivation above has been presented for the case of uninformative priors for and ; but it will be apparent that any priors that lead to a normal distribution being compounded with a scaled inverse chi-squared distribution will lead to a ''t''-distribution with scaling and shifting for , although the scaling parameter corresponding to above will then be influenced both by the prior information and the data, rather than just by the data as above.
Characterization
As the distribution of a test statistic
Student's ''t''-distribution with degrees of freedom can be defined as the distribution of the random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
''T'' with
:
where
* ''Z'' is a standard normal with expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
0 and variance 1;
* ''V'' has a chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
() with degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
;
* ''Z'' and ''V'' are independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
;
A different distribution is defined as that of the random variable defined, for a given constant ''μ'', by
:
This random variable has a noncentral ''t''-distribution with noncentrality parameter ''μ''. This distribution is important in studies of the power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
of Student's ''t''-test.
Derivation
Suppose ''X''1, ..., ''X''''n'' are independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
realizations of the normally-distributed, random variable ''X'', which has an expected value ''μ'' and variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
''σ''2. Let
:
be the sample mean, and
:
be an unbiased estimate of the variance from the sample. It can be shown that the random variable
:
has a chi-squared distribution with degrees of freedom (by Cochran's theorem). It is readily shown that the quantity
:
is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed with mean ''μ'' and variance ''σ''2/''n''. Moreover, it is possible to show that these two random variables (the normally distributed one ''Z'' and the chi-squared-distributed one ''V'') are independent. Consequently the pivotal quantity
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity nee ...
:
which differs from ''Z'' in that the exact standard deviation ''σ'' is replaced by the random variable ''S''''n'', has a Student's ''t''-distribution as defined above. Notice that the unknown population variance ''σ''2 does not appear in ''T'', since it was in both the numerator and the denominator, so it canceled. Gosset intuitively obtained the probability density function stated above, with equal to ''n'' − 1, and Fisher proved it in 1925.[
The distribution of the test statistic ''T'' depends on , but not ''μ'' or ''σ''; the lack of dependence on ''μ'' and ''σ'' is what makes the ''t''-distribution important in both theory and practice.
]
As a maximum entropy distribution
Student's ''t''-distribution is the maximum entropy probability distribution
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, ...
for a random variate ''X'' for which is fixed.
Properties
Moments
For , the raw moments of the ''t''-distribution are
:gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
to
:\operatorname E(T^k)= \nu^ \, \prod_^ \frac \qquad k\text,\quad 0
For a ''t''-distribution with \nu degrees of freedom, the expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
is 0 if \nu>1, and its variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
is \frac if \nu>2. The skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimo ...
is 0 if \nu > 3 and the excess kurtosis is \frac if \nu > 4.
Monte Carlo sampling
There are various approaches to constructing random samples from the Student's ''t''-distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value e ...
to uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
samples; e.g., in the multi-dimensional applications basis of copula-dependency. In the case of stand-alone sampling, an extension of the Box–Muller method and its polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
is easily deployed. It has the merit that it applies equally well to all real positive degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, ν, while many other candidate methods fail if ν is close to zero.[
]
Integral of Student's probability density function and ''p''-value
The function ''A''(''t'' , ''ν'') is the integral of Student's probability density function, ''f''(''t'') between −''t'' and ''t'', for ''t'' ≥ 0. It thus gives the probability that a value of ''t'' less than that calculated from observed data would occur by chance. Therefore, the function ''A''(''t'' , ''ν'') can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of ''t'' and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in ''t''-tests. For the statistic ''t'', with ''ν'' degrees of freedom, ''A''(''t'' , ''ν'') is the probability that ''t'' would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that ''t'' ≥ 0). It can be easily calculated from the cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
''F''''ν''(''t'') of the ''t''-distribution:
:A(t\mid\nu) = F_\nu(t) - F_\nu(-t) = 1 - I_\left(\frac,\frac\right),
where ''I''''x'' is the regularized incomplete beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t ...
(''a'', ''b'').
For statistical hypothesis testing this function is used to construct the ''p''-value.
Generalized Student's ''t''-distribution
In terms of scaling parameter ''σ̂'' or ''σ̂''2
Student's t distribution can be generalized to a three parameter location-scale family, introducing a location parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
\hat and a scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
\hat, through the relation
:X = \hat + \hat T
or
:T = \frac
This means that \frac has a classic Student's t distribution with \nu degrees of freedom.
The resulting non-standardized Student's ''t''-distribution has a density defined by:
:p(x\mid \nu,\hat,\hat) = \frac \left(1+\frac \left( \frac \right)^2\right)^
Here, \hat does ''not'' correspond to a standard deviation: it is not the standard deviation of the scaled ''t'' distribution, which may not even exist; nor is it the standard deviation of the underlying normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
, which is unknown. \hat simply sets the overall scaling of the distribution. In the Bayesian derivation of the marginal distribution of an unknown normal mean \hat above, \hat as used here corresponds to the quantity , where
:s^2 = \sum \frac.\,
Equivalently, the distribution can be written in terms of \hat^2, the square of this scale parameter:
:p(x\mid \nu, \hat, \hat^2) = \frac \left(1+\frac\frac\right)^
Other properties of this version of the distribution are:[
:\begin
\operatorname(X) &= \hat & \text \nu > 1 \\
\operatorname(X) &= \hat^2\frac & \text \nu > 2 \\
\operatorname(X) &= \hat
\end
This distribution results from ]compounding
In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
a Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
(normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
) with mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
\mu and unknown variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
, with an inverse gamma distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
placed over the variance with parameters a = \nu/2 and b = \nu\hat^2/2. In other words, the random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
''X'' is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is marginalized out (integrated out). The reason for the usefulness of this characterization is that the inverse gamma distribution is the conjugate prior
In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and t ...
distribution of the variance of a Gaussian distribution. As a result, the non-standardized Student's ''t''-distribution arises naturally in many Bayesian inference problems. See below.
Equivalently, this distribution results from compounding a Gaussian distribution with a scaled-inverse-chi-squared distribution with parameters \nu and \hat^2. The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. \nu = 2a, \; \hat^2 = \frac.
This version of the t-distribution can be useful in financial modeling. For example, Platen and Sidorowicz found that among the family of generalized hyperbolic distributions, this form of the t-distribution with about 4 degrees of freedom was the best fit for the ( log) return of many worldwide stock indices.
In terms of inverse scaling parameter ''λ''
An alternative parameterization
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, ...
in terms of an inverse scaling parameter \lambda (analogous to the way precision is the reciprocal of variance), defined by the relation \lambda = \frac\,. The density is then given by:
:p(x \mid \nu, \hat,\lambda) = \frac \left(\frac\right)^ \left(1+\frac\nu \right)^.
Other properties of this version of the distribution are:[
: \begin
\operatorname(X) &= \hat & & \text \nu > 1 \\ pt\operatorname(X) &= \frac\frac & & \text \nu > 2 \\ pt\operatorname(X) &= \hat
\end
This distribution results from ]compounding
In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
a Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
with mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
\hat and unknown precision (the reciprocal of the variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
), with a gamma distribution
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
placed over the precision with parameters a = \nu/2 and b = \nu/(2\lambda). In other words, the random variable ''X'' is assumed to have a normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
with an unknown precision distributed as gamma, and then this is marginalized over the gamma distribution.
Related distributions
* If X has a Student's ''t''-distribution with degree of freedom \nu then ''X''2 has an ''F''-distribution: X^2 \sim \mathrm\left(\nu_1 = 1, \nu_2 = \nu\right)
* The noncentral ''t''-distribution generalizes the ''t''-distribution to include a location parameter. Unlike the nonstandardized ''t''-distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
* The discrete Student's ''t''-distribution is defined by its probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
at ''r'' being proportional to: \prod_^k \frac \quad \quad r=\ldots, -1, 0, 1, \ldots . Here ''a'', ''b'', and ''k'' are parameters. This distribution arises from the construction of a system of discrete distributions similar to that of the Pearson distributions for continuous distributions.
* One can generate Student-''t'' samples by taking the ratio of variables from the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
and the square-root of . If we use instead of the normal distribution, e.g., the Irwin–Hall distribution
In probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a nu ...
, we obtain over-all a symmetric 4-parameter distribution, which includes the normal, the uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
, the triangular
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear, ...
, the Student-''t'' and the Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...
. This is also more flexible than some other symmetric generalizations of the normal distribution.
* ''t''-distribution is an instance of ratio distributions.
Bayesian inference: prior distribution for the degrees of the freedom
Suppose that x = (x_1,\cdots,x_N) represents N number of independently and identically distributed samples drawn from the Student t-distribution
t_(x) = \frac \left(1 + \frac \right)^, \quad x \in \mathbb.
With a choice a prior for the degrees of freedom \nu, denoted as \pi(\nu), Bayesian inference seeks to evaluate the posterior distribution
\pi(\nu, \textbf) = \frac, \quad \nu \in \mathbb^+.
Some popular choices of the priors are:
* Jeffreys prior
\pi_(\nu) \propto \left(\frac \right)^ \left( \psi'\left(\frac\right) -\psi'\left(\frac\right) -\frac\right)^,\quad \nu \in \mathbb^+,
where \psi'(x) represents trigamma function.
* Exponential prior
\pi_(\nu) =Ga(\nu, 1,0.1) = Exp(\nu, 0.1) = \frac e^,\quad \nu \in \mathbb^+
* Gamma prior
\pi_(\nu) =Ga(\nu, 2,0.1) =\frac e^,\quad \nu \in \mathbb^+
* Log-normal prior
\pi_(\nu) =logN(\nu, 1,1) =\frac \exp\left \frac \right\quad \nu \in \mathbb^+
The right panels show the result of the numerical experiments. The Bayes estimator based on the Jeffreys prior \pi_(\nu) results in relatively lower Mean Squared Error (MSE ) then the Maximum Likelihood Estimator (MLE) over the values \nu_0\in (0,25). It is important to note that no Bayes estimator dominates other estimators over the interval (0,25). In other words, each Bayes estimator has its own region where the estimator is non-inferior to others.
Uses
In frequentist statistical inference
Student's ''t''-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the ''t''-distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
would be used instead of the ''t''-distribution.
Confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
s and hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
s are two statistical procedures in which the quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile ...
s of the sampling distribution of a particular statistic (e.g. the standard score
In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the me ...
) are required. In any situation where this statistic is a linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
of the data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...
, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's ''t''-distribution. Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.
Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's ''t''-distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.
Hypothesis testing
A number of statistics can be shown to have ''t''-distributions for samples of moderate size under null hypotheses that are of interest, so that the ''t''-distribution forms the basis for significance tests. For example, the distribution of Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation ( statistical dependence betw ...
''ρ'', in the null case (zero correlation) is well approximated by the ''t'' distribution for sample sizes above about 20.
Confidence intervals
Suppose the number ''A'' is so chosen that
:\Pr(-A < T < A)=0.9,
when ''T'' has a ''t''-distribution with ''n'' − 1 degrees of freedom. By symmetry, this is the same as saying that ''A'' satisfies
:\Pr(T < A) = 0.95,
so ''A'' is the "95th percentile" of this probability distribution, or A=t_. Then
:\Pr \left (-A < \frac < A \right)=0.9,
and this is equivalent to
:\Pr\left(\overline_n - A \frac < \mu < \overline_n + A\frac\right) = 0.9.
Therefore, the interval whose endpoints are
:\overline_n\pm A\frac
is a 90% confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the ''t''-distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on a null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
.
It is this result that is used in the Student's ''t''-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the ''t''-distribution can be used to examine whether that difference can reasonably be supposed to be zero.
If the data are normally distributed, the one-sided (1 − ''α'')-upper confidence limit (UCL) of the mean, can be calculated using the following equation:
:\mathrm_ = \overline_n + t_\frac.
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, \overline_n being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL1−''α'' is equal to the confidence level 1 − ''α''.
Prediction intervals
The ''t''-distribution can be used to construct a prediction interval
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
for an unobserved sample from a normal distribution with unknown mean and variance.
In Bayesian statistics
The Student's ''t''-distribution, especially in its three-parameter (location-scale) version, arises frequently in Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
as a result of its connection with the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
. Whenever the variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
of a normally distributed random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
is unknown and a conjugate prior
In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and t ...
placed over it that follows an inverse gamma distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the resulting marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
of the variable will follow a Student's ''t''-distribution. Equivalent constructions with the same results involve a conjugate scaled-inverse-chi-squared distribution over the variance, or a conjugate gamma distribution over the precision. If an improper prior
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken int ...
proportional to ''σ''−2 is placed over the variance, the ''t''-distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a conjugate normally distributed prior, or is unknown distributed according to an improper constant prior.
Related situations that also produce a ''t''-distribution are:
* The marginal posterior distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
of the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
* The prior predictive distribution
Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be lowe ...
and posterior predictive distribution
Posterior may refer to:
* Posterior (anatomy), the end of an organism opposite to its head
** Buttocks, as a euphemism
* Posterior horn (disambiguation)
* Posterior probability, the conditional probability that is assigned when the relevant evi ...
of a new normally distributed data point when a series of independent identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
normally distributed data points have been observed, with prior mean and variance as in the above model.
Robust parametric modeling
The ''t''-distribution is often used as an alternative to the normal distribution as a model for data, which often has heavier tails than the normal distribution allows for; see e.g. Lange et al. The classical approach was to identify outliers
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter a ...
(e.g., using Grubbs's test) and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the ''t''-distribution is a natural choice of model for such data and provides a parametric approach to robust statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, suc ...
.
A Bayesian account can be found in Gelman et al. The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.
Student's ''t''-process
For practical regression and prediction
A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exac ...
needs, Student's ''t''-processes were introduced, that are generalisations of the Student ''t''-distributions for functions. A Student's ''t''-process is constructed from the Student ''t''-distributions like a Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...
is constructed from the Gaussian distributions
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
. For a Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...
, all sets of values have a multidimensional Gaussian distribution. Analogously, X(t) is a Student ''t''-process on an interval I=[a,b] if the correspondent values of the process X(t_1),...,X(t_n) (t_i \in I) have a joint Multivariate t-distribution, multivariate Student ''t''-distribution. These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi-output prediction, the multivariate Student ''t''-processes are introduced and used.
Table of selected values
The following table lists values for ''t''-distributions with ''ν'' degrees of freedom for a range of one-sided or two-sided critical regions. The first column is ''ν'', the percentages along the top are confidence levels, and the numbers in the body of the table are the t_ factors described in the section on #Confidence intervals, confidence intervals.
The last row with infinite ''ν'' gives critical points for a normal distribution since a ''t''-distribution with infinitely many degrees of freedom is a normal distribution. (See #Related distributions, Related distributions above).
Calculating the confidence interval
Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sided ''t''-value from the table is 1.372. Then with confidence interval calculated from
:\overline_n \pm t_\frac,
we determine that with 90% confidence we have a true mean lying below
:10 + 1.372 \frac = 10.585.
In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean.
And with 90% confidence we have a true mean lying above
:10 - 1.372 \frac = 9.414.
In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean.
So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval
:\left(10 - 1.372 \frac, 10 + 1.372 \frac\right) = (9.414, 10.585).
Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method; see confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
and prosecutor's fallacy.
Nowadays, statistical software, such as the R (programming language), R programming language, and functions available in many Spreadsheet, spreadsheet programs compute values of the ''t''-distribution and its inverse without tables.
See also
* ''F''-distribution
* Folded-t and half-t distributions, Folded-''t'' and half-''t'' distributions
* Hotelling's T-squared distribution, Hotelling's ''T''-squared distribution
* Multivariate Student distribution
* Standard normal table (''Z''-distribution table)
* t-statistic, ''t''-statistic
* Tau distribution, for internally studentized residuals
* Wilks' lambda distribution
* Wishart distribution
Notes
References
*
*
*
*
External links
*
Earliest Known Uses of Some of the Words of Mathematics (S)
''(Remarks on the history of the term "Student's distribution")''
* First Students on page 112.
Student's t-Distribution
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{{DEFAULTSORT:Student's T-Distribution
Continuous distributions
Special functions
Normal distribution
Compound probability distributions
Probability distributions with non-finite variance
Infinitely divisible probability distributions
Location-scale family probability distributions