A NALYSIS OF VARIANCE is used to provide the solution to two more or less distinct problems: (1) to detect and estimate components of variance in a composite population, (2) to detect and evaluate the significance of differences among means of sub-sets (Eisenhart, 1947). Attention has been drawn to complexities and some unsolved problems when there are disproportionate numbers of observations in each subclass of a multiple classification. But the case of proportionate subclass frequencies is usually passed over in a way which may lead the unwary to suppose that it has the same simplicity as when all sub-class means have equal weight. The purpose of this note is to call attention to the condition that such supposition is incorrect. When the sub-class numbers, although unequal, are proportionate to their marginal totals, it is well known that the additive property of sums of squares still holds good, and the analysis of variance can be carried through in the usual way. Beyond this it is generally implied although never explicitly stated, that interpretation as well as arithmetical procedure follows the usual lines; for example, Snedecor (1946), Sec. 11.9, writes, . . causing no injury to the analysis of variance. The implication however requires qualification (1) when the problem is to estimate components of variance, and (2) for tests of significance in which the within class variance fails to provide the appropriate estimate for error. Suppose that there are p classes A, A, 2 * , with numbers of observations proportional to a, , a2 ,. a, ; and q classes B1, B2 ...