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Range (statistics)
In descriptive statistics, the range of a set of data is size of the narrowest interval which contains all the data. It is calculated as the difference between the largest and smallest values (also known as the sample maximum and minimum). It is expressed in the same units as the data. The range provides an indication of statistical dispersion. Closely related alternative measures are the Interdecile range and the Interquartile range. Range of continuous IID random variables For ''n'' independent and identically distributed continuous random variables ''X''1, ''X''2, ..., ''X''''n'' with the cumulative distribution function G(''x'') and a probability density function g(''x''), let T denote the range of them, that is, T= max(''X''1, ''X''2, ..., ''X''''n'')- min(''X''1, ''X''2, ..., ''X''''n''). Distribution The range, T, has the cumulative distribution function ::F(t)= n \int_^\infty g(x) (x+t)-G(x) \, \textx. Gumbel notes that the "beauty of this formula is com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics) by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Julius Gumbel
Emil Julius Gumbel (18 July 1891, in Munich – 10 September 1966, in New York City) was a German mathematician and political writer. Gumbel specialised in mathematical statistics and, along with Leonard Tippett and Ronald Fisher, was instrumental in the development of extreme value theory, which has practical applications in many fields, including engineering and finance. In 1958, Gumbel published a key book, ''Statistics of Extremes'', in which he derived and analyzed the probability distribution that is now known as the Gumbel distribution in his honor. In the 1920s and early 1930s, Gumbel was considered unusual and highly controversial in German academic circles for his vocal support of left-wing politics and pacifism, and his opposition to Fascism. His influential writings about the politically motivated Feme murders made the case that the Weimar Republic was corruptly anti-leftist and anti-republican. Gumbel publicly opposed the Nazi Party and, in 1932, he was one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Deviation And Dispersion
Statistics (from German: ', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Range
In statistics, the studentized range, denoted ''q'', is the difference between the largest and smallest data in a sample normalized by the sample standard deviation. It is named after William Sealy Gosset (who wrote under the pseudonym "''Student''"), and was introduced by him in 1927. The concept was later discussed by Newman (1939), Keuls (1952), and John Tukey in some unpublished notes. Its statistical distribution is the '' studentized range distribution'', which is used for multiple comparison procedures, such as the single step procedure Tukey's range test, the Newman–Keuls method, and the Duncan's step down procedure, and establishing confidence intervals that are still valid after data snooping has occurred. Description The value of the studentized range, most often represented by the variable ''q'', can be defined based on a random sample ''x''1, ..., ''x''''n'' from the ''N''(0, 1) distribution of numbers, and another random variable ''s'' that is ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-estimator
In statistics, an L-estimator (or L-statistic) is an estimator which is a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median (of an odd number of values), or as many as all points, as in the mean. The main benefits of L-estimators are that they are often extremely simple, and often robust statistics: assuming sorted data, they are very easy to calculate and interpret, and are often resistant to outliers. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult. However, they are inefficient, and in modern times robust statistics M-estimators are preferred, although these are much more difficult computationally. In many circumstances L-estimators are reasonably efficient, and thus adequate for initial estimation. Examples A basic example is the median. Given ''n'' values x_1, \ldots, x_n, if n=2k+1 is odd, the median equals x_, the (n+1)/2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Statistic
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with Ranking (statistics), rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and non-parametric inference, inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other quantile, sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous probability distribution, continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution (continuous), uniform distribution. Notation and examples For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are :6, 9, 3, 7, the order statistics would be denoted : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Normal Distribution
In probability theory and 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 is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (statistics)
In statistics, interval estimation is the use of Sample (statistics), sample data to estimation, estimate an ''interval (mathematics), interval'' of possible values of a Statistical parameter, parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interval estimation are ''confidence intervals'' (a frequentism, frequentist method) and ''credible intervals'' (a Bayesian method). Less common forms include ''likelihood intervals,'' ''fiducial intervals,'' ''tolerance intervals,'' and ''prediction intervals''. For a non-statistical method, interval estimates can be deduced from fuzzy logic. Types Confidence intervals Confidence intervals are used to estimate the parameter of interest from a sampled data set, commonly the mean or standard deviation. A confidence interval states there is a 100γ% confidence that the parameter of interest is within a lower and upper bound. A common misconception of confidence interva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Every probability distribution Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
