One-Way ANOVA: Comparing Three or More Groups

To compare the means of two groups we use a t-test. But what if we have three, four, or more groups? This is where one-way ANOVA (Analysis of Variance) comes in.

Why not multiple t-tests?

Running several pairwise t-tests inflates the Type I error rate. With $\alpha = 0.05$ per test, the chance of finding at least one false significant result across three comparisons climbs to about 14%.

The F statistic

ANOVA compares between-group variance to within-group variance:

$$F = \frac{\text{Between-group variance}}{\text{Within-group variance}} = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

When the differences between groups are large relative to the random variation within groups, $F$ grows and the $p$ value shrinks.

Assumptions

A quick look with Python

from scipy import stats

group_a = [82, 85, 88, 90, 86]
group_b = [78, 81, 84, 80, 83]
group_c = [91, 94, 89, 95, 92]

f_stat, p_value = stats.f_oneway(group_a, group_b, group_c)
print(f"F = {f_stat:.2f}, p = {p_value:.4f}")

When you find a significant result ($p < 0.05$), move on to post-hoc tests (such as Tukey HSD) to see which groups differ.

Run it in Basira

Try the steps above on your own data in one click — assumption checks, post-hoc tests, and effect size are computed automatically.