# Contingency Table Chi-Square Fisher Exact Cramér V Tutor Step-by-step independence testing for 2×2 / R×C contingency tables: Pearson χ², Yates, G-test, Fisher exact, effect sizes, adjusted residuals, Cochran rule and optional Chapman estimate. > Canonical page: https://elysiatools.com/en/tools/contingency-table-chi-square-fisher-exact-cramer-v-tutor - **Category:** Data Analysis - **Keywords:** chi square, fisher exact, contingency table, cramers v, yates, g test, adjusted residual ## Overview χ² = Σ(O−E)²/E; Yates for 2×2; G = 2ΣO·ln(O/E); p via regularized incomplete gamma; Fisher via lgamma hypergeometric with SciPy two-tailed convention; Cramér's V / phi / C; Haberman residuals; BH-FDR note; Chapman N̂ for 2×2 mark-recapture. ## Inputs - **Contingency table (rows on lines, cells comma/space/tab-separated)** (textarea): 30, 20 20, 30 - **Yates continuity correction** (select) - **Significance level α** (number) - **Chapman capture-recapture estimate (2×2)** (checkbox): For 2×2 mark-recapture data: N̂ = (a+1)(b+1)/(c+1) − 1. ## When to use - Test whether two categorical variables are statistically independent. - Use Fisher's exact test when a 2×2 table contains small expected counts. - Measure association strength and identify cells that contribute most to a table-level result. ## How it works - Enter the contingency table with rows on separate lines and values separated by commas, spaces, or tabs. - The tool calculates expected counts, Pearson χ², degrees of freedom, p-value, and the G-test. - For 2×2 tables, choose whether Yates continuity correction is applied automatically, always, or never. - Review Fisher's exact result, effect sizes, adjusted residuals, Cochran guidance, and the optional Chapman estimate when applicable. ## Use cases - Compare outcomes across treatment groups, survey categories, or demographic segments. - Evaluate small-sample categorical experiments where exact testing is more appropriate than an asymptotic chi-square result. - Inspect adjusted residuals to locate the cells behind a significant association. ## Frequently asked questions ### What table formats are supported? The tool accepts 2×2 and R×C tables, with rows on separate lines and cells separated by commas, spaces, or tabs. ### When is Fisher's exact test recommended? It is recommended for 2×2 tables when expected counts are small and the chi-square approximation may be unreliable. ### What does the significance level α control? Alpha sets the threshold for judging statistical significance, with a default of 0.05. ### What does Cramér's V measure? Cramér's V summarizes the strength of association between categorical variables. ### What is the Chapman estimate used for? 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