Sum of Squares

The sum of squares (SS) is the fundamental quantity that an ANOVA table decomposes: the total squared deviation of observations from a reference value, typically the mean. It is the arithmetic engine of variance analysis, transforming raw differences into a metric that can be partitioned, compared, and tested. The total sum of squares is split into components — between groups, within groups, and interaction terms — each carrying a portion of the overall variation.

Despite its centrality, the sum of squares is a deceptively simple measure. It treats all deviations as equally important, regardless of direction or context, and it amplifies large deviations quadratically. A single outlier can dominate the sum of squares, making the decomposition sensitive to distributional assumptions that are rarely satisfied in practice. The sum of squares is not a discovery; it is a convention — one chosen for mathematical convenience in the era of hand calculation, not for epistemic clarity. Its persistence in modern software is a case study in how computational inertia shapes scientific practice. The mean square error refines the sum of squares by dividing by degrees of freedom, but the underlying assumption that squared deviations are the right metric remains unexamined.