Wikipedia: Student's t-distribution

Student's t-distribution
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Student was the pseudonym of William Sealey Gosset (1876-1937), who, in 1908, published a paper (cited below) showing that a certain probability distribution, now conventionally called Student's distribution or the t-distribution, arises in the problem of estimating the mean of a normally distributed population when the sample size is small.

(Perhaps a "pure" mathematician could say "... when the sample size is small and the standard deviation is unknown and has to be estimated from the data." In practice the standard deviation of the population is always unknown and must be estimated from the data. Textbook problems treating the standard deviation as if it were known are of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance 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.)

Suppose X₁, ..., X_n are independent random variables that are normally distributed with expected value μ and variance σ². Let

be the "sample mean", and

be the "sample variance". It is readily shown that

is normally distributed with mean 0 and variance 1. Gosset studied a related quantity,

and showed that T has the probability density function

with ν equal to n − 1. The distribution of T is now called the t-distribution. The parameter ν is conventionally called the number of degrees of freedom. The distribution depends ν, 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.

A more general result can be derived. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let W have a normal distribution with mean 0 and variance 1. Let V have a chi-square distribution with ν degrees of freedom. Then the ratio

has a t-distribution with ν degrees of freedom.

The expected value of the t-distribution is 0, and its variance is (n − 1)/(n − 3). The cumulative distribution function is given by an incomplete beta function,

with

The interval whose endpoints are

where A is an appropriate percentage-point of the t-distribution, is a confidence interval for μ.

The overall shape of the probability density function of the t-distribution 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. The t-distribution is related to the F-distribution as follows: the square of a value of t with n degrees of freedom is distributed as F with 1 and n degrees of freedom.

References

"Student" (W.S. Gosset) (1908) The probable error of a mean. Biometrika 6(1):1--25. Available on-line through http://www.jstor.com
M. Abramowitz and I. A. Stegun, eds. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. See Section 26.7.
R.V. Hogg and A.T. Craig (1978) Introduction to Mathematical Statistics. New York: Macmillan.