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One-way analysis of variance -- compare means across multiple groups with F-test, effect sizes, ANOVA table, and Python code.
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Quick Reference:
Effect Size (η²):
One-way ANOVA assumes: (1) independence of observations, (2) normality within each group (less critical for large samples by CLT), (3) homogeneity of variances (equal variances across groups). If variances are unequal, consider Welch's ANOVA. For non-normal data, consider the Kruskal-Wallis test as a non-parametric alternative.
Use ANOVA when comparing 3 or more groups. Running multiple t-tests inflates the family-wise Type I error rate. For example, with 4 groups, you would need 6 pairwise t-tests, and the probability of at least one false positive at alpha=0.05 rises to ~26%. ANOVA controls this by testing all groups simultaneously with a single F-test.
A significant ANOVA tells you at least one group differs but not which. Common post-hoc tests include: Tukey HSD (controls family-wise error, good for all pairwise comparisons), Bonferroni (conservative, flexible), Scheffé (most conservative, for complex contrasts), and Games-Howell (when variances are unequal). Tukey HSD is the most commonly used.
Eta squared (η² = SSB/SST) is the proportion of total variance explained by group membership. Guidelines: 0.01 = small, 0.06 = medium, 0.14 = large. Omega squared (ω²) is a less biased estimator, especially for small samples. It adjusts for the positive bias in η² and is generally preferred in research reports.