ANOVA: Analysis of Variance

What is ANOVA?

ANOVA tests whether the means of three or more groups are significantly different. Instead of running multiple t-tests (which inflates error), ANOVA tests them all at once.

The key idea

ANOVA compares two types of variation:

• Between-group variation: How much do the group means differ from each other?
• Within-group variation: How much do values within each group vary?

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

If $F$ is large → the groups are significantly different.

Step-by-step

Groups: A = {85, 90, 88}, B = {78, 82, 80}, C = {92, 95, 88}

Step 1 — Group means: $\bar{A} = 87.67$, $\bar{B} = 80$, $\bar{C} = 91.67$. Overall mean $\bar{X} = 86.44$.

Step 2 — Between-group SS: $SS_B = 3[(87.67-86.44)^2 + (80-86.44)^2 + (91.67-86.44)^2] = 3[1.51 + 41.47 + 27.30] = 210.89$

Step 3 — Within-group SS: Compute deviations within each group: $SS_W = (4.67+5.43+0.11) + (4+4+0) + (0.11+11.09+13.47) = 42.89$

Step 4 — F-statistic: $df_B = 2$, $df_W = 6$.

$F = \frac{SS_B/df_B}{SS_W/df_W} = \frac{210.89/2}{42.89/6} = \frac{105.44}{7.15} = 14.75$

Step 5 — Critical value at $\alpha = 0.05$: $F_{2,6} = 5.14$.

$14.75 > 5.14$ → Reject $H_0$. The teaching methods produce significantly different results.

Try it in the visualization

Three groups of data points are displayed. The between-group and within-group variations are shown visually. The F-statistic is computed and compared to the critical value.