Enter the sum of squares between groups and the degrees of freedom into the Calculator. The calculator will evaluate the Mean Sum of Squares Between Groups.
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MSB Formula
The mean square between groups (MSB) measures the average variation among group means in a one-way ANOVA. It is the between-groups sum of squares divided by its degrees of freedom.
MSB = SSB / (k - 1)
- MSB = mean square between groups
- SSB = sum of squares between groups
- k = number of groups
- k - 1 = degrees of freedom between groups (df between)
When you start from group statistics rather than a finished ANOVA table, SSB is computed from group sizes and means against the grand mean:
SSB = Σ n_i * (x̄_i - x̄_grand)^2
- n_i = sample size of group i
- x̄_i = mean of group i
- x̄_grand = weighted mean of all observations
To run the full F test, the calculator also computes within-group mean square and the F ratio:
MSW = SSW / (N - k) F = MSB / MSW
- SSW = sum of squares within groups
- N = total number of observations across all groups
- N - k = degrees of freedom within groups (df within)
Assumptions for the F test: independent samples, approximately normal residuals within each group, and roughly equal group variances. The MSB value itself does not require these assumptions, but its interpretation as a variance estimate under the null hypothesis does.
How each calculator mode maps to the formulas:
- Quick MSB applies MSB = SSB / df between directly. Use it when you already have both numbers.
- ANOVA From Sums takes SSB, SSW, and both df values, then returns MSB, MSW, F, p, F critical, η², and ω².
- From Summary Stats rebuilds SSB and SSW from each group's n, mean, and sample SD using SSW = Σ (n_i - 1) * s_i².
- From Raw Data computes group means, sample SDs, SSB, and SSW from raw values you paste in.
Reference Tables
Typical F critical values for the between-groups test at α = 0.05. Read df between across the top and df within down the side.
| df within \ df between | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 |
| 15 | 4.54 | 3.68 | 3.29 | 3.06 | 2.90 |
| 20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 |
| 30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 |
| 60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 |
| 120 | 3.92 | 3.07 | 2.68 | 2.45 | 2.29 |
Effect size benchmarks for one-way ANOVA based on η² (eta squared):
| η² value | Effect size | Interpretation |
|---|---|---|
| 0.01 to under 0.06 | Small | About 1 to 5 percent of variance explained by group. |
| 0.06 to under 0.14 | Medium | Group differences explain a noticeable share of variance. |
| 0.14 and above | Large | Group membership explains a substantial share of variance. |
Worked Example and FAQ
Example. Three teaching methods are tested with 10 students each (N = 30, k = 3). The ANOVA table reports SSB = 136.1 and SSW = 451.4.
- df between = k - 1 = 2
- df within = N - k = 27
- MSB = 136.1 / 2 = 68.05
- MSW = 451.4 / 27 = 16.72
- F = 68.05 / 16.72 = 4.07
F critical at α = 0.05 with df 2 and 27 is about 3.35. Since 4.07 is greater than 3.35, you reject the null hypothesis that all three population means are equal.
What does MSB actually represent? It is an estimate of the population variance based only on differences between group means. If the null hypothesis is true, MSB and MSW estimate the same variance, so their ratio F should be near 1.
Why divide SSB by k - 1 instead of k? Once you fix the grand mean, only k - 1 of the group means are free to vary. Dividing by the degrees of freedom rather than the count yields an unbiased variance estimate under the null.
Can MSB be larger than MSW with no real effect? Yes. Random sampling can make MSB exceed MSW even when group means are equal in the population. The F distribution accounts for that, which is why you compare F to a critical value rather than just checking if MSB is bigger.
What if my groups have different sample sizes? The formulas still apply. The SSB calculation weights each squared deviation by n_i, and df within becomes N - k where N is the total across all groups.
Is MSB the same as variance between groups? It is the unbiased estimate of the common variance assuming the null hypothesis. The raw variance of the group means (without weighting by n) is a different quantity and is not used directly in the F test.
