ChiQuadratTest

ChiQuadratTest, commonly referred to as the chi-squared test, is a statistical hypothesis test used for categorical data to assess how well observed frequencies match expected frequencies under a null hypothesis. It is employed in two main settings: goodness-of-fit tests, where a single categorical distribution is compared to a specified model, and tests of independence, where the relationship between two categorical variables is evaluated in a contingency table.

The test statistic is X^2 = sum over all categories of (O_i − E_i)^2 / E_i, where O_i are observed counts and E_i are expected counts under the null hypothesis. The chi-squared distribution is used to approximate the sampling distribution of X^2 for sufficiently large sample sizes. Degrees of freedom depend on the context: for goodness-of-fit, df = k − 1 with k categories; for a contingency table with r rows and c columns, df = (r − 1)(c − 1). A p-value derived from the chi-squared distribution indicates whether to reject the null hypothesis at a chosen significance level.

Key assumptions include independent observations, mutually exclusive categories, and adequately large expected counts (commonly E_i ≥ 5). The test is approximate; for small samples or many cells with low expected counts, Fisher’s exact test or exact methods may be preferred. Variants include the Yates continuity correction for 2×2 tables and the likelihood-ratio chi-squared statistic. Effect size measures such as Cramér’s V or the phi coefficient provide a sense of practical association. ChiQuadratTest is implemented in statistical software under functions such as chi-squared tests for goodness-of-fit and tests of independence.