# Student's Distribution

## Student's distribution

[′stüd·əns ‚dis·trə′byü·shən]*n*observations comes from a normal population with a given mean.

## Student’s Distribution

(or *t* distribution). Student’s distribution with *f* degrees of freedom is the distribution of the ratio *t = X/Y* of two independent random variables *X* and *Y* such that *X* is normally distributed with mathematical expectation *EX =* 0 and variance *DX* = 1 and *fY*^{2} has a chi-square distribution with *f* degrees of freedom.

The distribution function for *t* is

Let *X*_{1}, ..., *X*_{n} be independent random variables with the same normal distribution. If E*X _{i}*, =

*a*and D

*X*= σ

_{i}^{2}(

*i*= 1,...,

*n*), then, for any real values of

*a*and for

*σ*> 0, the ratio has a Student’s distribution with

*f*=

*n*– 1 degrees of freedom; here,

*X̄*= ∑

*X*and

_{i}/n*s*

^{2}= ∑(

*X*–

_{i}*X̄*)/(

*n*– 1). This property of the ratio was first made use of in 1908 by the British statistician W. Gosset to solve an important problem in classical error theory. The problem involves the testing of the hypothesis

*a*=

*a*

_{0}, where

*a*

_{0}is a given number and the variance σ

^{2}is assumed to be unknown. If the inequality

is satisfied, the hypothesis *a* = *a*_{0} is regarded as not contradicting the observation results *X*_{1}..., *X _{n}*. If the inequality does not hold, the hypothesis is rejected. Such a test involving the use of

*t*is known as Student’s test, or the

*t*test. The critical value

*t*=

*t*(α) is the solution of the equation

_{n–1}*S*= 1 - α/2, where α is a specified significance level (0 < α < ½). If the hypothesis

_{n–1}(t)*a*=

*a*

_{0}is true, then a is the probability that the hypothesis will be rejected when the test is performed with the critical value

*t*(α).

_{n–1}Student’s distribution is used in solving many other problems in mathematical statistics (*see*SAMPLE, SMALL; ERRORS, THEORY OF; and LEAST SQUARES, METHOD OF).

### REFERENCE

Cramér, H.*Matematicheskie metody statistiki*, 2nd ed. Moscow, 1975. (Translated from English.)