Overview
Chi-square tests are frequently applied to data that violates their assumptions — particularly the minimum expected frequency requirement. When any cell in a contingency table has an expected frequency below 5, the chi-square approximation is unreliable; Fisher's exact test must be used instead. More critically, a significant chi-square result tells the researcher only that there is some relationship between the categorical variables — not where in the table the relationship is concentrated. Standardized residuals are required to identify the specific cells that deviate from independence.
The Chi-Square Testing Framework selects the appropriate test variant, verifies expected frequency assumptions, calculates Cramér's V for effect size, and interprets standardized residuals to locate the specific categorical patterns driving statistical significance.