A Note on Karl Pearson’s 1900 Chi-Squared Test: Two Derivations of the Asymptotic Distribution, and Uses in Goodness of Fit and Contingency Tests of Independence, and a Comparison with the Exact Sample Variance Chi-Square Result

Karl Pearson’s chi-squared test is widely known and used, both as a goodness-of-fit test for hypothesized distributions or frequencies, and in tests of independence in contingency tables. The test was introduced in Pearson (1900), but the derivation in that paper is almost incomprehensible. Two derivations of the asymptotic distribution are given here. The first uses joint characteristic functions, and the second uses a multivariate central limit theorem. Goodness-of-fit tests and contingency table tests of independence are discussed, and the asymptotic chi-square distribution result for Pearson’s test statistic is compared and contrasted with the exact chi-square result for the sample variance estimator.