In textbooks on multivariate statistics, the topic of multivariate analysis of variance (MANOVA) is usually presented in terms of the decomposition of the “total sums of squares and products” matrix into the “within” and “between” matrices, often called the “hypothesis” and the “error” matrices. While this decomposition can be justified by maximum likelihood estimation and hypothesis testing under normality assumptions, better motivation is provided by a finite mixture model in which no assumptions beyond the existence of second moments are needed. We propose that the decomposition be interpreted in terms of estimates of conditional and unconditional moments, rather than as just an algebraic identity.