The best-known situations in which the chi-square distribution is used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

The chi-square distribution is a special case of the gamma distribution.

If X1, …, Xk are independent, standard normal random variables, then the sum of their squares

is distributed according to the chi-square distribution with k degrees of freedom. This is usually denoted as

The chi-square distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i.e. the number of Xi’s)

Further properties of the chi-square distribution can be found in the box at right.

The probability density function (pdf) of the chi-square distribution is

where Γ(k/2) denotes the Gamma function, which has closed-form values at the half-integers.

For derivations of the pdf in the cases of one and two degrees of freedom, see Proofs related to chi-square distribution.

Its cumulative distribution function is: