SSwithin

SSwithin, short for within-group sum of squares, is a component of the total sum of squares used in analysis of variance (ANOVA). It measures how much the observations vary within each group or treatment level, as opposed to variation between groups captured by SSbetween. In a one-way ANOVA with k groups, each group i contains n_i observations x_ij (j = 1..n_i), and x̄_i is the mean of group i, while x̄ is the grand mean. The within-group sum of squares is SSwithin = sum_{i=1}^k sum_{j=1}^{n_i} (x_ij − x̄_i)^2. The totals decompose as SSTotal = SSbetween + SSwithin, where SSbetween = sum_i n_i (x̄_i − x̄)^2. Degrees of freedom: SSwithin has df = N − k, where N = sum_i n_i. Interpretation: SSwithin reflects experimental error or residual variability after accounting for group means; smaller SSwithin indicates more homogeneity within groups. In most ANOVA computations, SSwithin is used to estimate the mean square error (MSE) for tests of group effects, with MSE = SSwithin / dfWithin. In software outputs, SSwithin is often labeled SSE or SSW, depending on the convention. In repeated-measures or within-subject designs, analogous within-subject sums of squares exist to capture variability within the same subject across conditions.