Are two categorical variables associated?
A contingency table crosses two classifications: study method by pass/fail, coffee preference by age band. The test of independence asks whether the row variable and the column variable move together, or whether the column mix is the same inside every row, as independence demands. Expected counts are built from the margins alone: E = row total × column total ÷ N.
Three revision methods against two outcomes (pass/fail counts 30/20, 25/25, 15/30; N = 140): χ² = 6.836 on 2 df, p = 0.0328. Method and outcome are associated at the 0.05 level. Cramér's V = 0.217, a small-to-moderate association: real, but method explains only part of who passes. The observed-vs-expected table shows where: method 3 fails more students than independence predicts (30 observed vs 21.4 expected).
Association is not causation
A significant chi-square certifies pattern, not mechanism. Method 3's students may differ before they revise: weaker intake, evening jobs. Observational tables cannot untangle that; random assignment can. Report the association and Cramér's V honestly and let the design carry the causal claims.
Small expected counts in a 2×2 table. The chi-square approximation frays below expected counts of about 5: Fisher's exact test is the instrument built for that corner.
Frequently asked questions
What does Cramér's V add?
Chi-square grows with sample size even for weak associations. V rescales it to 0–1 (roughly: 0.1 small, 0.3 medium, 0.5 large), so strength can be compared across tables and studies.
How do I enter a 2x2 table?
Two lines, two counts each: e.g. '64, 56' then '48, 62'. The calculator handles the rest.