Suppose `X`_{1}, ..., `X`_{n} are independent random variables that are normally distributed with expected value μ and variance `σ ^{2}`. Let

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.

- "Student" (W.S. Gosset) (1908) The probable error of a mean.
*Biometrika*6(1):1--25. Available on-line through " class="external">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.