This note has as its primary purpose the correction of a statement made in a previous paper regarding the attainment of the variance of a proposed estimator of its lower bound. At the same time it seemed useful, as an expository function, to recapitulate a development of the lower bound which led this writer to a better understanding of it, and to the realization of the indicated error. Here the bound is developedab initiofrom a simple identity, instead of secondarily from Schwartz's inequality, as is usual. It should be stated that, mathematically, there is nothing essentially new in this development; however, it yields, in addition to a clarification of the “why” of the bound, a necessary, condition for its attainment. The element of novelty in the statement of this condition is doubtless mathematically trivial, but may be of use in application. Also it is hoped that its direct derivation from the same elementary identity as the bound itself is derived, may make it understood by the nonmathematician—and possibly even by mathematicians! The usual estimate of the mean of the normal function, whose variance is well known to attain the lower bound, is used as an example to illustrate how this condition is fulfilled, and the logistic function illustrates how it is not fulfilled.