Chi-Squared Distribution

Probability · Axiom Academy

LESSON Chi-Squared Distribution The distribution of sums of squared standard normals 1. Definition and Construction The chi-squared distribution with k degrees of freedom is the distribution of the sum Z₁² + Z₂² + ... + Z_k² where each Z_i ~ N(0,1). The degrees of freedom parameter k determines the shape of the distribution. 2. Shape Varies with Degrees of Freedom For small k (like 1 or 2), the distribution is highly right-skewed. As k increases, the distribution becomes more symmetric and bell-shaped, eventually approaching normality by the Central Limit Theorem. The chi-squared distribution is used in hypothesis tests for variance, goodness-of-fit tests, and tests of independence in contingency tables. The sample variance of normally distributed data follows a scaled chi-squared distribution.

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