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 fY² has a chi-square distribution with f degrees of freedom.

The distribution function for t is

Let X₁, ..., X_n be independent random variables with the same normal distribution. If EX_i, = a and DX_i = σ² (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_i/n and s² = ∑(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 = a₀, where a₀ is a given number and the variance σ² is assumed to be unknown. If the inequality

is satisfied, the hypothesis a = a₀ is regarded as not contradicting the observation results X₁..., 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_n–1 (α) is the solution of the equation S_n–1(t) = 1 - α/2, where α is a specified significance level (0 < α < ½). If the hypothesis a = a₀ is true, then a is the probability that the hypothesis will be rejected when the test is performed with the critical value t_n–1(α).

Student’s distribution is used in solving many other problems in mathematical statistics (seeSAMPLE, SMALL; ERRORS, THEORY OF; and LEAST SQUARES, METHOD OF).