Compare Three or More Group Means With ANOVA

While you could compare multiple groups using pairwise t-tests, this approach inflates the Type I error rate (false positives). With 3 groups at α = 0.05, three t-tests give a family-wise error rate of about 14% instead of 5%. ANOVA controls this by testing all groups simultaneously in a single test. If ANOVA is significant, you then use post-hoc tests (like Tukey or Bonferroni) that correct for multiple comparisons.

A significant ANOVA (p < α) tells you that at least one group mean is significantly different from the others, but it doesn't tell you which specific groups differ. To identify which pairs of groups are significantly different, you need to perform post-hoc tests. Common choices include Tukey's HSD, Bonferroni correction, or Scheffé's test.

Eta squared represents the proportion of total variance in the dependent variable explained by the independent variable (group membership). Cohen's guidelines suggest: η² < 0.01 is negligible, 0.01-0.06 is small, 0.06-0.14 is medium, and > 0.14 is large. For example, η² = 0.25 means 25% of the total variance in scores can be attributed to group differences.

If normality is violated, ANOVA is fairly robust with sample sizes ≥ 15-20 per group. For severe non-normality, use the Kruskal-Wallis test (non-parametric alternative). If homogeneity of variance is violated (test with Levene's test), use Welch's ANOVA, which doesn't assume equal variances. With very small or unequal sample sizes, consider bootstrapping or permutation tests.

Technically yes, and the result will be equivalent to an independent samples t-test (F = t²). However, the t-test is more commonly used for two groups as it's more intuitive and allows for directional (one-tailed) hypotheses. ANOVA is specifically designed for comparing three or more groups and is most useful in that context.

One-way ANOVA has one independent variable (factor) with multiple levels (groups). Two-way ANOVA has two independent variables and can test for main effects of each factor plus their interaction. For example, one-way might test effects of different drugs, while two-way could test drugs × dosage, examining whether drug effects depend on dosage level.

While there's no strict minimum, general guidelines suggest at least 5-10 observations per group for basic analyses, with 15-20+ preferred for robust results. Equal sample sizes provide the most statistical power and make ANOVA more robust to violations of homogeneity of variance. Use power analysis to determine appropriate sample sizes based on expected effect size.

The F-distribution is the sampling distribution of the ratio of two independent chi-squared variables divided by their degrees of freedom. In ANOVA, it represents the distribution of the F-statistic under the null hypothesis (no group differences). The F-distribution is always right-skewed and non-negative, with shape determined by the two degrees of freedom values.

ANOVA tests whether any group means differ (omnibus test), not the direction of differences. The F-statistic is always positive (ratio of variances), and only large F values indicate significant differences. Small F values (near 1) suggest group means are similar. There's no meaningful 'left tail' because you can't have negative variance ratios.

A complete ANOVA report should include: (1) F-statistic and degrees of freedom: F(df_between, df_within), (2) p-value, (3) effect size (η² or partial η²), (4) descriptive statistics per group (n, mean, SD), (5) whether homogeneity assumptions were met, and (6) post-hoc test results if significant. Example: 'F(2, 45) = 8.34, p = 0.001, η² = 0.27. Tukey's HSD showed Group A differed significantly from Groups B and C.'