In probability and statistics, Student's t-distribution (or simply the t-distribution) is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.
However, has heavier tails and the amount of probability mass in the tails is controlled by the parameter . For the Student's t distribution becomes the standard Cauchy distribution, whereas for it becomes the standard normal distribution .
The Student's t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.
In the form of the location-scale t-distribution it generalizes the normal distribution and also arises in the Bayesian analysis of data from a normal family as a compound distribution when marginalizing over the variance parameter.
History and etymology
In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper.
In the English-language literature, the distribution takes its name from William Sealy Gosset's 1908 paper in Biometrika 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 in Dublin, Ireland, 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, who called the distribution "Student's distribution" and represented the test value with the letter t.
Definition
Probability density function
Student's t-distribution has the probability density function (PDF) given by
where is the number of degrees of freedom and is the gamma function. This may also be written as
where B is the Beta function. In particular for integer valued degrees of freedom we have:
For even,
For odd,
The probability density function is symmetric, 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 (CDF) can be written in terms of I, the regularized incomplete beta function. For t > 0,
where
Other values would be obtained by symmetry. An alternative formula, valid for , is
where 2F1 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.
Moments
For , the raw moments of the t-distribution are
Moments of order or higher do not exist.
The term for , k even, may be simplified using the properties of the gamma function to
For a t-distribution with degrees of freedom, the expected value is 0 if , and its variance is if . The skewness is 0 if and the excess kurtosis is if .
Location-scale t-distribution
Location-scale transformation
Student's t-distribution generalizes to the three parameter location-scale t-distribution by introducing a location parameter and a scale parameter . With
and location-scale family transformation
we get
The resulting distribution is also called the non-standardized Student's t-distribution.
Density and first two moments
The location-scale t distribution has a density defined by: