ANOVA F-Statistic Calculator Formula
Understand the math behind the anova f-statistic calculator. Each variable explained with a worked example.
Formulas Used
F-Statistic
f_stat = ms_between / ms_withindf Between
df_between = k - 1df Within
df_within = n_total - kEta-Squared (Effect Size)
eta_squared = ms_between * (k - 1) / (ms_between * (k - 1) + ms_within * (n_total - k))Variables
| Variable | Description | Default |
|---|---|---|
ms_between | Mean Square Between Groups (MSB) | 120 |
ms_within | Mean Square Within Groups (MSW) | 30 |
k | Number of Groups (k) | 4 |
n_total | Total Sample Size (N) | 40 |
How It Works
One-Way ANOVA F-Test
ANOVA tests whether the means of two or more groups are significantly different by comparing between-group variance to within-group variance.
Formula
F = MSB / MSW
where MSB = SSB/(k-1) is the mean square between groups and MSW = SSW/(N-k) is the mean square within groups. A large F indicates that group means differ more than expected by random chance.
Worked Example
MSB = 120, MSW = 30, k = 4 groups, N = 40 total observations.
- 01F = 120 / 30 = 4.000
- 02df1 = 4 - 1 = 3, df2 = 40 - 4 = 36
- 03Critical F(3,36) at alpha=0.05 ≈ 2.87
- 04F = 4.0 > 2.87, so groups differ significantly
Frequently Asked Questions
What assumptions does ANOVA require?
Independence of observations, normality within each group (checked with Shapiro-Wilk test), and homogeneity of variances (checked with Levene test). ANOVA is robust to mild violations with equal group sizes.
What do I do after a significant F-test?
A significant F-test says at least two means differ, but not which ones. Use post-hoc tests (Tukey HSD, Bonferroni, Scheffe) for pairwise comparisons while controlling family-wise error rate.
What is eta-squared?
Eta-squared (eta²) = SSB/SST is the proportion of total variance explained by group membership. It is the ANOVA analog of R². Cohen guidelines: 0.01 small, 0.06 medium, 0.14 large effect.
Ready to run the numbers?
Open ANOVA F-Statistic Calculator