Enter the group sizes (number of scores in each group), the mean score for each group, and (optionally) the grand mean into the calculator to determine the sum of squares between groups.

SS Between Calculator

Enter group sizes and group means as comma-separated lists (one value per group). If you leave GM blank, it will be calculated as the weighted mean.

Related Calculators

SS Between Formula

SS Between, also called the sum of squares between groups, measures how far each group mean is from the grand mean after weighting each group by its size. In one-way ANOVA, this is the part of total variability explained by differences between groups.

SSB = \sum_{i=1}^{k} n_i(M_i - GM)^2

If the grand mean is not entered manually, it should be computed as the weighted mean of the group means.

GM = \frac{\sum_{i=1}^{k} n_iM_i}{\sum_{i=1}^{k} n_i}

What each term means

SSB
Sum of squares between groups.
ni
Size of group i, or the number of observations in that group.
Mi
Mean of group i.
GM
Grand mean across all observations from all groups.
k
Total number of groups.

How to calculate SS Between

  1. List the size of each group and the mean for each group.
  2. Find the grand mean. If group sizes are different, use the weighted grand mean.
  3. For each group, subtract the grand mean from the group mean.
  4. Square each difference.
  5. Multiply each squared difference by that group’s size.
  6. Add all group contributions together.

Key interpretation points

  • Larger SSB means the group means are more spread out around the grand mean.
  • SSB is always nonnegative because squared deviations cannot be negative.
  • SSB equals 0 only when every group mean is exactly the same as the grand mean.
  • Group size matters because larger groups contribute more to the total between-group variation.
  • The units are squared, so SSB is measured in squared units of the original variable.

Why the grand mean should usually be weighted

When group sizes are unequal, the grand mean is not the simple average of the group means. Each group mean must be weighted by its number of observations. If all groups have the same size, the weighted and unweighted grand means are the same.

Example

Suppose there are three groups with sizes 8, 10, and 6, and their means are 72, 81, and 75.

GM = \frac{8(72) + 10(81) + 6(75)}{8 + 10 + 6} = \frac{1836}{24} = 76.5
SSB = 8(72 - 76.5)^2 + 10(81 - 76.5)^2 + 6(75 - 76.5)^2
SSB = 8(20.25) + 10(20.25) + 6(2.25) = 162 + 202.5 + 13.5 = 378

So the sum of squares between groups is 378.

How SS Between fits into ANOVA

In one-way ANOVA, total variability is split into between-group variation and within-group variation.

SST = SSB + SSW

SS Between is then converted into a mean square by dividing by its degrees of freedom.

df_{between} = k - 1
MSB = \frac{SSB}{k - 1}

The ANOVA F statistic compares between-group variation to within-group variation.

F = \frac{MSB}{MSW}

This means SSB by itself does not determine statistical significance. It must be interpreted together with within-group variability.

Useful special case

If every group has the same number of observations, the formula simplifies because the common group size can be factored out.

SSB = n\sum_{i=1}^{k}(M_i - GM)^2

Common input mistakes

  • Entering a different number of group sizes and group means.
  • Using a simple average of the group means instead of the weighted grand mean when group sizes are unequal.
  • Entering raw scores instead of group means.
  • Using negative or zero group sizes.
  • Assuming a large SSB automatically means the ANOVA result is significant.

Tips for using the calculator

  • Enter group sizes as a comma-separated list, with one size for each group.
  • Enter group means in the same order as the group sizes.
  • Leave the grand mean blank if you want it calculated automatically from the group sizes and means.
  • If you enter the grand mean manually, make sure it matches the full dataset structure.
  • Decimals are fine for means, but group sizes should be whole numbers.