ChiQuadrat

ChiQuadrat, commonly written chi-square distribution (χ²-distribution), is a continuous probability distribution with k degrees of freedom, denoted χ²(k). It arises as the distribution of the sum of squares of k independent standard normal random variables.

Mathematically, if Z1,...,Zk are independent N(0,1) variables and X = Σ Zi², then X ~ χ²(k). The probability density function is f(x; k) = 1/(2^{k/2} Γ(k/2)) x^{k/2 − 1} e^{−x/2} for x > 0. The distribution is a special case of the gamma distribution with shape k/2 and scale 2, i.e., χ²(k) ~ Gamma(k/2, 2).

Moments of the central chi-square distribution are E[X] = k and Var[X] = 2k, with skewness √(8/k). The noncentral chi-square distribution χ²(k, λ) extends the model with noncentrality parameter λ, having mean k + λ and variance 2(k + 2λ).

Applications: the chi-square distribution underpins Pearson’s chi-square statistic χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, used in goodness-of-fit tests and tests of independence in contingency tables. Under the null hypothesis, χ² follows a central χ²(k) distribution, where k is the number of categories minus one minus any estimated parameters. It also appears in confidence intervals, model assessments, and as an approximation in likelihood-ratio tests for large samples.

Limitations and considerations: reliable use requires adequate expected frequencies (commonly at least 5 per cell). For small samples, exact tests or simulations may be preferable. The distribution is related to the gamma family and to normal limits for large degrees of freedom.