ANOVA

Analysis of variance — comparing means across three or more groups by partitioning total variability into between-group and within-group components.

Diagram · Group Distributions

Drag the sliders to move group means. ANOVA asks: is the spread between groups larger than the spread within groups?

Group 1: 2.0

Group 2: 5.0

Group 3: 8.0

Between-group SS: 180.0Within-group SS: 43.2F = 56.25 → reject H₀

Definition

ANOVA (Analysis of Variance) tests whether the means of three or more groups are all equal.

$H_0$ : $\mu_1 = \mu_2 = \cdots = \mu_k$ (all group means are equal)

$H_1$ : at least one mean differs

ANOVA decomposes total variability in data into:

Between-group variance: how much group means vary from the grand mean
Within-group variance: how much individual observations vary within their group

If between-group variance is much larger than within-group variance, we have evidence against $H_0$ .

Key properties

Total sum of squares always splits exactly into between-group and within-group components — nothing is lost or double-counted
Under $H_0$ , the F-statistic has expectation close to 1, since both numerator and denominator estimate the same noise variance
A significant ANOVA result only establishes that some mean differs, not which pairs
Assumes normality within groups, equal variances across groups, and independence of observations

Common mistakes

Running multiple pairwise t-tests instead of post-hoc corrected tests: this reintroduces the multiple-testing problem that ANOVA's single omnibus test was meant to avoid
Ignoring unequal variances across groups: violating homoscedasticity distorts the F-test's Type I error rate — check with Levene's test, or use Welch's ANOVA as a robust alternative

Three fertilizers

Three fertilizers applied to 5 plots each. Crop yields (kg): A: 23, B: 31, C: 23.

The grand mean is 24.6. Group means differ noticeably. ANOVA tests if these differences are larger than expected from random variation.

Try it

ANOVA uses the F-statistic $F = \text{MS}_{\text{between}} / \text{MS}_{\text{within}}$ . Why is F the right ratio to use? When would you expect F to be large vs. close to 1?

Solution

Under $H_0$ , both $\text{MS}_{\text{between}}$ and $\text{MS}_{\text{within}}$ estimate the same population variance $\sigma^2$ , so their ratio is close to 1.

When $H_0$ is false (means differ), the between-group variance picks up the signal (differences between means), while the within-group variance still only estimates noise. So $F$ grows large. A large $F$ is evidence against $H_0$ .

Related concepts

Statistics· Statistical Tests

T-TestTesting whether a mean differs from a reference value or whether two group means differ, using the t-distribution when sample sizes are small.

Statistics· Descriptive

VarianceThe average squared distance of each value from the mean — a measure of spread.

Statistics· Statistical Tests

F-TestComparing two variances using the ratio of sample variances, which follows the F-distribution under the null hypothesis of equal population variances.

Statistics· Statistical Tests

Multiple TestingThe inflation of false positives when running many tests simultaneously — and corrections like Bonferroni and the false discovery rate.