Student was the pseudonym of William Sealey Gosset (1876-1937), who, in 1908, published a paper (citation 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. 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.)
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
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,
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.
(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