If Z1, Z2,…, Zν are ν independent standard normal variables (see normal distribution), and if Y is defined by
then Y has a chi-squared distribution with ν degrees of freedom (written as
). The probability density function f is given by
where Γ is the gamma function. The form of the distribution was first given by Abbe in 1863 and was independently derived by Helmert in 1875 and Karl Pearson in 1900. It was Pearson who gave the distribution its current name.
The chi-squared distribution has mean ν and variance 2ν. For ν≤2 the mode is at 0; otherwise it is at (ν−2). A chi-squared distribution is a special case of a gamma distribution. The case ν=2 corresponds to the exponential distribution. Percentage points for chi-squared distributions are given in Appendix VIII.