The relation between item difficulty distributions and the “validity” and reliability of tests is computed through use of normal correlation surfaces for varying numbers of items and varying degrees of item intercorrelations. Optimal or near optimal item difficulty distributions are thus identified for various possible item difficulty distributions. The results indicate that, if a test is of conventional length, is homogeneous as to content, and has a symmetrical distribution of item difficulties, correlation with a normally distributed perfect measure of the attribute common to the items does not vary appreciably with variation in the item difficulty distribution. Greater variation was evident in correlation with a second duplicate test (reliability). The general implications of these findings and their particular significance for evaluating techniques aimed at increasing reliability are considered.

A rapid method of estimating a correlation coefficient is given. The method expresses the correlation coefficient as the ratio between two differences in sums (or means) of the dependent variable computed only for extremes of the bivariate distribution. A trial shows that this method gives results similar to the product-moment correlation coefficient. Extensions of the method to qualitative data are also suggested.

This paper presents a prepunched deck of cards to enable the extraction of square roots on standard punch card tabulating equipment. Such a deck is valuable in constructing mathematical tables which involve square roots or in obtaining standard deviations in connection with computing correlation coefficients. By using a deck of reciprocals in conjunction with the deck for square roots, correlations may be solved completely on IBM equipment.

A theoretical discussion of the factor pattern of predictor tests and criterion shows that ordinary test selection methods break down under certain circumstances. It is shown that maximal results may not occur if suppressor variables are present among the predictors. Suggested solutions to the problem include: (1) prior item analysis of tests against the criterion, (2) selection of several trial batteries including some with suppressor variables on the basis of a factor analysis of tests and criterion, (3) modification of the usual test selection procedures to include separate solutions based upon each of several starting variables, or (4) the cumbersome and tedious solution of all possible combinations of predictors. The solutions are recommended in the order named above. Although all of the suggested solutions involve added labor and may not be necessary, the test or battery constructor should at least be aware of the problem.

A comparison of factorial analyses made by Thurstone and by the writer shows that they differ in three important respects. The results of the two analyses are compared with respect to their social utility, which is offered as a proper criterion for judging the merit of factorial analyses performed by different mathematical procedures.