This article deals with a factorial analysis of Thurstone's Primary Mental Abilities Tests. The analysis was made in order to determine whether the tests designated to measure a particular ability would be found upon completion of the analysis to be clustered together with significant loadings and be isolated from the tests of the other abilities. The results show that most of the tests specifically designed to measure six of the abilities were isolated with loadings varying in size. The tests for the remaining ability were not isolated. It is also shown that the tests are complex in nature, measuring more than any one single ability.

The mathematical derivation of a test for determining the fiducial limits of, and significance of difference between, means when the samples are drawn from exponential populations is presented. The test for differences between means takes the particularly simple form of the F test (the ratio of the larger to the smaller mean) with each mean possessed of 2n degrees of freedom, n being the number of cases in the sample. Random sampling, a range of scores upwards from a lower limit of zero, and independence of means from each other are necessary assumptions for the use of the test. Examples of situations in which the test should be used are given, together with a description of the necessary computational procedures. Comparisons of the results of the application of this test with the erroneous application of the critical ratio on actual data show that rather large discrepancies exist between the two tests. Results obtained by applying tests which assume normality to exponential distributions are subject to much error.

A general formula for the reliability of a weighted composite has been derived by which that reliability can be estimated from a knowledge of the weights whatever their source, reliabilities, dispersions, and intercorrelations of the components. The Spearman-Brown formula has been shown to be a special case of the general statement. The effect of the internal consistency or intercorrelation of the components has been investigated and the conditions defining the set of weights yielding maximum reliability shown to be that the weight of a component is proportional to the sum of its intercorrelations with the remaining components and inversely proportional to its error variance.

Given a battery of n tests that has already been resolved into r orthogonal common factors and n unique factors, procedures are outlined for computing the following types of linear multiple regressions directly from the factor loadings: (i) the regression of any one test on the n−1 remaining tests; (ii) all the n different regressions of order n−1 for the n tests, computed simultaneously; (iii) the regression of any common factor on the n tests; (iv) the regressions of all the common factors on the n tests computed simultaneously; (v) the regression of any unique factor on the n tests; (vi) the regressions of all the unique factors on the n tests, computed simultaneously. Multiple and partial correlations are then determined by ordinary formulas from the regression coefficients. A worksheet with explicit instructions is provided, with a completely worked out example. Computing these regressions directly from the factor loadings is a labor-saving device, the efficiency of which increases as the number of tests increases. The amount of labor depends essentially on the number of common factors. This is in contrast to computations based on the original test intercorrelations, where the amount of labor increases more than proportionately as the number of tests increases. The procedures evaluate formulas developed in a previous paper (2). They are based essentially on a shortened way of computing the inverse of the test intercorrelation matrix by use of the factor loadings.

A matching method proposed by Dr. C. E. Stuart is presented in some detail and the essentials for a test of significance are derived. This method differs from the older matching methods in that partial credit is allowed for a near miss. A slight variation of the method permits the matching of one item with M sets of n traits.

Tables are given for σr√N for the tetrachoric correlation coefficient for the following values of the correlation in the population: .00, ± .10, ± .20, ..., ± .80, ± .90, ± .95.