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One-way Analysis of Variance (ANOVA) tests whether the means of three or more groups are all equal, using H₀: μ₁ = μ₂ = … = μₖ. The F-statistic is the ratio of between-group variance to within-group variance: F = MS_between / MS_within. A large F suggests means differ more than would be expected from chance alone. ANOVA does not identify which specific means differ — post-hoc tests (like Tukey's HSD) are required for pairwise comparisons after rejecting H₀.
Run a simple experiment: measure plant heights under three different fertilizers. Partition total variability into between-group and within-group components in an ANOVA table. Emphasize why running multiple t-tests inflates the Type I error rate — this motivates ANOVA as the correct approach.
You already know the two-sample t-test: compare two group means by asking how many standard errors separate them. One-way ANOVA extends this logic to three or more groups. The natural instinct is to run all pairwise t-tests — with five groups you'd run ten — but this has a serious problem you've encountered in hypothesis testing: Type I error inflation. If each test has a 5% false-positive rate and tests are independent, the probability of at least one false positive across ten tests climbs toward 40%. ANOVA provides a single test that handles all groups simultaneously, preserving the overall error rate at α.
The central idea is a decomposition of variance. Take all the observations together and measure their total variation around the grand mean — the overall mean of all groups combined. This total sum of squares splits into two additive pieces: SS_between, which measures how far each group mean sits from the grand mean (weighted by group size), and SS_within, which measures how much individual observations scatter around their own group mean. SS_within is the baseline noise — variation that cannot be explained by group membership. SS_between is the signal — variation attributable to the groups themselves.
The F-statistic is the ratio F = MS_between / MS_within, where MS (mean square) divides each sum of squares by its degrees of freedom to make the quantities comparable. MS_between uses df = k − 1 (k groups), and MS_within uses df = N − k (N total observations). Under the null hypothesis that all group means are equal, both MS_between and MS_within estimate the same population variance σ², so F should be near 1. When group means genuinely differ, MS_between inflates while MS_within remains anchored to within-group noise, pushing F above 1. The F-distribution gives the probability of observing a ratio this large by chance alone.
A significant F-test tells you "at least one group mean differs," not which ones. Post-hoc tests like Tukey's Honestly Significant Difference (HSD) perform all pairwise comparisons with a correction that controls the familywise error rate at α — solving the multiple-comparison problem that motivated ANOVA in the first place. The procedure assumes roughly normal data within groups and approximately equal variances (homoscedasticity). The normality assumption is fairly robust for moderate sample sizes by the central limit theorem, but unequal variances can distort the F-test; in that case, Welch's ANOVA provides a correction analogous to Welch's t-test for the two-group setting.