A stimulus (or stimulus-complex) is pictured as giving rise to a random series of sensory nerve “pulses,” which manifest themselves in contractions of individual muscle fibers. Assuming the expected time-frequency of these pulses to be proportional to the intensity of the stimulus, probability distributions are computed representing the cumulative effect of these pulses on the state of the organism, that is, on its degree of awareness of the stimulus. Preliminary results suggest a modification of the Weber-Fechner formula for intensity discrimination for certain types of stimuli: the psychological scale to be measured by I1/2 instead of log I.

An analysis by Thurstone's centroid method of the intercorrelations of fifty-two tests was carried to ten factors. Included were tests of social intelligence, Philip's attention tests, and Seashore's tests of musical ability. After rotation of axes, the most important factors appeared to pertain to operations conventionally alluded to by the following terms: verbal facility; spatial ability; numerical ability; attention; musical ability; and memory (or memory span). The social intelligence tests proved to be mainly tests of the verbal factor. A factorial sex difference was indicated by the superiority of men in tests of spatial ability.

A shortened method of finding the regression for estimation of the scores on mental factors is presented. If the correlations among n tests are accounted for by r factors, the regression equation involves the inverse of an r × r matrix instead of the inverse of an n × n matrix. When r is much less than n, there is considerable saving in computational labor. A numerical example is presented to illustrate the method.

The results of J. P. Guilford's card-sorting experiment by the method of equal appearing intervals are analyzed mathematically. The constants of the psychometric functions are obtained by processes analogous to the constant process. When these constants are determined (a) independent of the supposition that the series of groups into which the cards were sorted is quantitative in character and (b) under this supposition, good agreement is found between them. Guilford's results agree with Weber's and Fechner's laws in both cases.

Certain assumptions and procedures basic to factor analysis are examined from the point of view of the mathematician. It is demonstrated that the Hotelling method does not yield meaningful traits, and an example from the theory of gas mixtures with convertible components is cited as evidence. The justification of current methods for determining the adequacy of the reproduction of a correlation matrix by a factorial matrix is questioned, and a x2 criterion, practical only for a small matrix, is proposed. By means of a hypothetical example from geometry, it is shown that results of a Hotelling analysis are necessarily relative to the population at hand. The factorial effects of the adjunction of a “total test” to a group of tests are considered. Some of the general considerations and questions raised are pertinent to types of analysis other than the Hotelling.

A rigorous and an approximate solution are found for the problem: Given a primary trait matrix for n tests and r1 traits, and a matrix for the same n tests and r2 reference axes, to discover the transformation which will transform the second matrix into the first, or primary trait matrix. Formulas for determining the limits of the effect of using the approximate solution are presented. The method is applied to a set of twenty hypothetical tests, defined by their loadings on four orthogonal primary traits. After factoring the inter-correlations of these variables by Thurstone's centroid method, approximating the diagonals, the original hypothetical matrix is reproduced with a root mean square discrepancy of .014 by assuming as known the primary trait loadings of only the first eight tests. The method is applied to the results of factoring two batteries of 14 tests, having 8 tests in common, to give the factor loadings of the two batteries on the same reference axes. The method provides a means of comparing directly and quantitatively the results of two different factor studies, provided they have tests in common, and of testing the stability of simple structure under changes in the battery. The relations of the method here developed to certain problems in multiple correlation are shown.

A method is indicated by which multiple factor analysis may be used in determining a number, r, and then in selecting r “predicting” variables out of n variables so that each of the remaining n-r variables may be predicted almost as well from the r variables as it could be predicted from all the n−1 variables.