chi-squared distribution

Variant: χ²

If Z₁, Z₂,..., Z_ν are ν independent standard normal variables, and if Y is defined by

, then Y has a chi-squared distribution with ν degrees of freedom (written as χ_ν²). 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 X.


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.

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