ANOVA F-Test Formula
The ANOVA F-test compares means of three or more groups using F = MS_between / MS_within to determine if group differences are significant.
The Formula
ANOVA (Analysis of Variance) tests whether the means of three or more groups are significantly different. Instead of running multiple t-tests (which increases error risk), ANOVA performs a single test using the F-statistic.
The F-statistic compares the variance between group means to the variance within the groups. A large F value suggests that the group means differ more than you would expect by random chance.
Variables
| Symbol | Meaning |
|---|---|
| F | F-statistic (test value to compare against critical F) |
| MS_between | Mean square between groups = SS_between / df_between |
| MS_within | Mean square within groups = SS_within / df_within |
| SS_between | Sum of squares between groups: Σnᵢ(x̄ᵢ − x̄)² |
| SS_within | Sum of squares within groups: ΣΣ(xᵢⱼ − x̄ᵢ)² |
| df_between | k − 1 (where k = number of groups) |
| df_within | N − k (where N = total number of observations) |
Example 1
Three teaching methods are tested on groups of 5 students each. Group means: Method A = 78, Method B = 85, Method C = 72. Grand mean = 78.33. SS_between = 422.2, SS_within = 300. Is there a significant difference at α = 0.05?
df_between = k − 1 = 3 − 1 = 2
df_within = N − k = 15 − 3 = 12
MS_between = 422.2 / 2 = 211.1
MS_within = 300 / 12 = 25.0
F = 211.1 / 25.0 = 8.44
Critical F(2, 12) at α = 0.05 ≈ 3.89
F = 8.44 > 3.89, so we reject the null hypothesis. The teaching methods produce significantly different results.
Example 2
Four fertilizer types are tested on crop yield (kg). Each group has 6 plots. SS_between = 120, SS_within = 480. Is there a significant effect?
df_between = 4 − 1 = 3
df_within = 24 − 4 = 20
MS_between = 120 / 3 = 40.0
MS_within = 480 / 20 = 24.0
F = 40.0 / 24.0 = 1.67
Critical F(3, 20) at α = 0.05 ≈ 3.10
F = 1.67 < 3.10, so we fail to reject the null hypothesis. No significant difference between fertilizers.
When to Use It
Use ANOVA when comparing means across three or more groups.
- Comparing treatment effects in medical or agricultural experiments
- Testing whether different manufacturing processes produce different results
- Education research: comparing test scores across teaching methods
- Marketing: comparing customer satisfaction across product variants
- Any experiment with one categorical independent variable and a continuous dependent variable
Note: ANOVA tells you that at least one group differs, but not which one. Use post-hoc tests (like Tukey's HSD) to identify which specific groups differ.