One question, three or more means
With two groups you run a t-test. With three or more, running every pairwise t-test inflates the false-positive rate with each extra comparison. Three groups mean three tests, and a 5% error rate per test quietly becomes about 14% per family. One-way ANOVA asks the joint question first: is the variation between group means larger than the variation inside groups can explain?
Three classes, same quiz. Means: 84.6, 75.6 and 90.0 (five students each). Between-groups SS = 529.2 (df 2, MS = 264.6); within-groups SS = 186.4 (df 12, MS = 15.53). F = 17.03, p = 0.0003. The group means are far too spread out for the within-class noise to explain. η² = 0.740, about 74% of all variation in marks is between-class variation.
What a significant F does not say
It does not say which classes differ, or by how much. ANOVA is a gatekeeper: pass it, and pairwise follow-ups (with a correction such as Tukey's HSD) earn their turn. Fail it, and pairwise fishing is exactly the false-positive machine ANOVA exists to prevent.
Assumptions, in plain sight
- Independent observations, within and between groups.
- Roughly normal residuals in each group. ANOVA tolerates mild departures, especially with balanced group sizes.
- Similar variances across groups. Wildly different spreads argue for Welch's ANOVA instead.
Reporting "the ANOVA was significant, so class 3 beats class 1" without a post-hoc test. F alone never names the pair.
Frequently asked questions
Can ANOVA handle two groups?
Yes, it reduces exactly to the pooled two-sample t-test (F = t², same p). There is just no reason to prefer it there.
What is eta squared?
The share of total variation attributable to group membership: SS_between ÷ SS_total. It is ANOVA's effect size: report it next to the p-value.