Exact procedures for determining the expected values of sample mean squares in terms of population parameters are adequately described in a number of places in sytatistical literature (1.3.7). For simmple designs with few classifications the processes can be gone through quickly, and with practice, the expectations of such mean squares can be written by inspection. However, when a design involves several classifications, and particularly when the classifications are a mixture of random and fixed varieties, the process become complex and tedious. The purpose of this paper is to illustrated a set of simple rules which reduces the processes of determining the expectations of the mean squares of even complex analyses to practically the equivalent of determination by inspection by inspection. These rules are sufficiently general to cover all complexities of classification, provide the sums or means at each level of summarization are composed of equal numbers of observations and, in the case of random varieties, are drawn from infinite populations.

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