**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***Student's distribution**or the

**, 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.)**

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

*T*has the probability density function

*n*− 1. The distribution of

*T*is now called the

**. The parameter ν is conventionally called the "degrees of freedom". The distribution depends ν, but not μ or σ; the lack of dependence on μ and σ is what makes the**

`t`-distribution*t*-distribution important in both theory and practice.

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

*t*-distribution with ν degrees of freedom.

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,

- ,

- .

`A`is an appropriate percentage-point of the

`t`-distribution, is a confidence interval for μ.

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.

### References

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