Student's t-distribution

"Student" was the pseudonym of William Sealey Gosset (1876-1937), who, in 1908, published a pseudonymous paper 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 would 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 as if it were certain, and (2) those that illustrate mathematical reasoning; 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

<math>\overline{X}_n=(X_1+\cdots+X_n)/n</math>

be the "sample mean", and

<math>S_n^2=\frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2</math>

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

<math>Z=\frac{\overline{X}_n-\mu}{\sigma/\sqrt{n}}</math>

is normally distributed with mean 0 and variance 1. Student found the probability distribution of

<math>T=\frac{\overline{X}_n-\mu}{S_n/\sqrt{n}};</math>

that distribution is "the t-distribution with n-1 degrees of freedom."

Clearly this distribution does not depend on the values of μ or σ. Its expected value is 0 and its variance is (n-1)/(n-3).

The interval whose endpoints are

<math>\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}</math>

where A is an appropriate percentage-point of the t-distribution, is a confidence interval[?] for μ. The formula for the probability density function of the t-distribution with n-1 degrees of freedom is known and these confidence intervals can therefore be readily computed once the sample mean and sample variance have been determined.

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.