A type of non-negative continuous probability distribution, normally written as the χ2-distribution, with one parameter ν called the degrees of freedom. The distribution (see appendix 15) is skewed to the right and has the property that the sum of independent random variables each having a χ2-distribution also has a χ2-distribution. It is used in the chi-squared test for measuring goodness of fit, in tests on variance and in testing for independence in contingency tables. It has mean ν and variance 2ν.
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.
Chi-squared distribution. All chi-squared distributions have ranges from 0 to ∞. Their shape is determined by the value of ν. If ν>2 then the distribution has a mode at (ν−2); otherwise the mode is at 0.
An important probability distribution with many uses in statistical analysis. Denoted by the Greek symbol Χ2, it is the distribution of the sum of squares of f independent random variables, each being drawn from the normal distribution with zero mean and unit variance. The integer f is the number of degrees of freedom. Critical values of the probability distribution are widely available in tables, but exact calculations involve the incomplete gamma function. The most common applications are
(a) testing for interactions between different classifications of data using contingency tables;
(b) testing goodness-of-fit;
(c) forming confidence intervals for estimates of variance.