Chi-Square Tests

Chi-Square Tests

Overview

Chi-square tests address two fundamental questions about categorical data. The goodness of fit test determines whether a single categorical variable follows a specified distribution (e.g., is a die fair?). The test of independence determines whether two categorical variables are associated (e.g., is gender associated with voting preference?). Both use the same test statistic — the sum of squared differences between observed and expected frequencies, standardized by expected frequencies. When expected frequencies are small (), the chi-square approximation breaks down and Fisher's exact test should be used instead. Effect size is measured by Cramér's V.

Key Concepts

Degrees of Freedom

Minimum Expected Frequency Rule

• No expected frequency should be less than 1
• No more than 20% of expected frequencies should be less than 5
• For tables violating this, use Fisher's exact test

Cramér's V Benchmarks

Quick Example

Key Takeaways

Deep Dive

For detailed explanations, worked examples, and Python implementations, explore the dedicated statistics lessons:

Chi-Square Distribution

• Chi-Square Distribution — PDF, properties, relationship to Normal, critical values, and simulation

Goodness of Fit

• Chi-Square Goodness of Fit — Testing whether data matches a specified distribution with full worked examples

Test of Independence

• Chi-Square Test of Independence — Contingency tables, expected frequencies, post-hoc analysis, and standardized residuals

Related Tests

• F-Test for Equality of Variances — Comparing variances of two groups using the F-distribution
• Levene's Test — Robust alternative to F-test for equality of variances

Related Topics

• Hypothesis Testing — The framework underlying chi-square tests
• Nonparametric Methods — Distribution-free alternatives when assumptions are violated
• Logistic Regression — Models the same associations chi-square detects, with continuous predictors