SSBetween

SSbetween, or the sum of squares between groups, is a component of the analysis of variance (ANOVA) that measures the variability of group means relative to the overall grand mean. It captures how much the group means differ from the overall mean, reflecting effects of the grouping factor.

Mathematically, SSbetween is calculated as the sum over all groups of the group size times the squared difference between each group mean and the grand mean: SSbetween = sum_j n_j (mean_j − grand_mean)^2. If group sizes are equal (n), this simplifies to n times the sum across groups of (mean_j − grand_mean)^2.

SSbetween is one part of the total variability, with SSTotal = SSbetween + SSwithin (also called SSresidual or SSerror). The degrees of freedom associated with SSbetween is k − 1, where k is the number of groups. The mean square for between groups is MSbetween = SSbetween / (k − 1). The F statistic used to test whether all group means are equal is F = MSbetween / MSwithin, where MSwithin is the mean square within groups.

Interpretation focuses on the relative size of SSbetween compared with SSwithin. A larger SSbetween suggests greater differences among group means, leading to a larger F statistic if the within-group variability is not also large. However, conclusions depend on the F-test and underlying assumptions (independence, normality, and homogeneity of variances).

Notes: SSbetween is central to one-way ANOVA and is commonly used in between-subjects designs. Related terms include SSwithin, SSTotal, grand mean, and degrees of freedom.