In probability theory and statistics , the chi-square distribution (also chi-squared or χ² distribution) is one of the most widely used theoretical probability distributions in inferential statistics , i.e. in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

If X_i are k independent , normally distributed random variables with mean 0 and variance 1, then the random variable

is distributed according to the chi-square distribution. This is usually written

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

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

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 . However, many other statistical tests lead to a use of this distribution.