It is assumed that a battery of n tests has been resolved into components in a common factor space of r dimensions and a unique factor space of at most n dimensions, where r is much less than n. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's “single factor” prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of test j on the remaining n−1 tests to approach the communality of test j as a limit asn increases indefinitely while r remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.

When the centroid method of factor analysis was applied to two sets of data on athletic performances, three significant factors emerged: strength, velocity, and dead weight. Scores on this speed factor were predicted by the multiple regression technique, the factor loadings on the speed factor being used as the criterion correlations, and these predicted scores were correlated with each of the other variables. When the original tables, augmented by the new speed variable, were refactored, the computed speed factor fell on the speed axis as a primary trait. It is thus shown that it is possible to isolate and measure a factor which appears in variables under consideration only as a compound.

If we may take the habits, attitudes and utterances of people of low mentality and little independence and originality as evidence of a primitive type of thinking and those of thoughtful people as evidence of a more highly developed type, we can expect to find in comparative sociology the mental distinctions that characterize primitive and more highly developed peoples. If we compare the initial stages of a mental discipline with the later stages we can find evidence of the same distinctions. If we examine the mental processes of everyday life which require little effort and which we can engage in in indolent moods and compare with those that are distinctly effortful and which, as we perform them, carry the thrill of awakeness and vitality, we again find evidence of the same distinctions. In each of these comparisons the more primitive thought is largely in terms of broad categories and its decisions are unambiguous, unqualified and lead to unquestioned conclusion, while the higher type is either definitely quantitative or concerned with much finer categories and leads to qualified conviction and tempered judgment.

The intercorrelations of thirty-seven variables, including the Minnesota battery of “mechanical ability” tests, the seven MacQuarrie tests of “mechanical ability,” O'Connor's Wiggly blocks, and the Stenquist picture-matching test, were analyzed by Thurstone's centroid method. Five factors, Perceptual, Verbal, Youth, Manual Agility, and Spatial, were taken out. Factors prominent in so-called mechanical ability tests are the Spatial and Perceptual ones with Mac-Quarrie's dotting test significantly high in the Manual Agility factor. Each of the factors can be measured with group pencil-and-paper tests.

A modification of the method of successive intervals is presented which yields scale values correlating .995 with those from Thurstone's method described by Saffir. Values yielded by the present method can be obtained in 25 per cent of the time required by the older method and are shown to be, on a priori grounds, more reliable as well.

In failing to define the units in which the stimulus is to be measured, the Weber law might seem to make no definite assertion, and indeed, it is shown that any single empirical function, supposed to relate a given stimulus intensity with that intensity which is just noticeably greater, can be put into the Weber form by a suitable change of scale in which the stimulus intensity is to be measured. Nevertheless, it turns out that if different individuals have different Weber functions, when the intensities are measured on a given scale, then it is by no means always possible to transform the scale so that all of the functions can take on the Weber form. Some necessary conditions are given for the possibility of such a transformation when there is at hand a finite number of functions, and when the functions depend upon a single parameter the necessary and sufficient condition is easily derived. The same discussion leads to a generalization of Thurstone's psychophysical scale and shows that such a scale is always possible.

The relations of abilities, as measured by Thurstone's Tests for Primary Mental Abilities, to activity preferences, as measured by Kuder's Preference Record, are investigated for a population of 512 university freshmen. Ability profiles for contrasted groups on each preference scale reveal relatively slight overlapping between the two sets of measures, although the apparent trends are reasonable. The Pearson inter-correlation coefficients of all pairs of measures involved were determined. Implications of the findings in relation to theory and to educational and vocational guidance are indicated.

Tests or personal inventories with differential item response weights may be scored by means of punch card equipment. Detailed instructions are given for preparing the cards and scoring the forms. The scoring speed is approximately four to eight times that attained by manual scoring.

Delinquent boys are compared with non-delinquents with respect to their attitudes towards a series of “good” and “bad” social acts, by the use of scales having rational origins of measurement. A new technique, essentially an extension of Thurstone's Method of Successive Intervals, is found to give results similar to Horst's Method of Balanced Values. Significant differences in mean attitude between the two groups are not found.

A new approximation method for obtaining a maximized multiple criterion, based on the formula of Edgerton and Kolbe (2), is presented. By applications to examples from the literature (1, 2), the new method is evaluated in comparison with the Horst approximation (1), a suggested revision of the Horst procedure, and the more exact but more laborious iterative method for the principal axis solution of Hotelling.

In continuation of a previous study, somewhat more complex relations are considered for the case of interaction of two classes, of which one “organizes” the activities of the other. By way of illustration it is shown how different assumptions concerning the attitudes of the individuals to such an interaction of classes lead to different mathematical expressions.

In the application of the analysis of variance to data obtained in educational methods experiments which involve several classes of several schools, one assumption is that of homogeneity in the variances of pupil scores from school to school. It is shown that such variances on representative educational achievement tests are heterogeneous. The effects of this heterogeneity upon the F-tests of significance commonly employed in methods experiments are investigated by comparing the actual distribution of F values for a large number of “experiments” involving marked heterogeneity with a theoretical distribution based on the assumption of homogeneity. Although the findings, which vary somewhat with the type of variance ratio, are not entirely conclusive, they apparently demonstrate that departure from homogeneity does not invalidate the use of the customary F-tests for evaluating results of the typical methods experiment.

This paper shows how to compute multiple correlation coefficients, partial correlation coefficients, and regression coefficients from the factorial matrix. Special emphasis is given to computation technique and to approximation formulas. The method is extremely flexible in application since it may be applied to any subset of the original set of observed variables. It is also extremely useful when many of these coefficients are desired.

The study of William Brown and William Stephenson, “A Test of the Theory of Two Factors,” is re-analyzed by means of the Thurstone multiple factor methods. No tests or correlations are left out of the original table of correlations as is done in the original analysis in an attempt to validate the two-factor theory. Space, verbal, and perceptual speed factors similar to those found by Thurstone, Wright, and Garrett are identified. A common factor of “Maturation” is postulated to account for the remaining communality of the tests. A fifth factor is considered to have no significance due to the small amount of variance which it contributes to the total.

It is shown that invariance requirements remove the indeterminacy in factor determination and lead to an integration of factorial studies with promise of considerable reduction in computational labor. The selection of significant primary factors is discussed, with special reference to Thurstone's simple structure criterion.

An analytic technique for the study of trait variability is presented. An expression for the average variance from test to test and an expression for the variance of these variances are derived in terms of the number of tests and the intercorrelations between them, and limiting cases are examined. The question of the true relationship between the nature of the distribution of test scores in a sample of N persons and the nature of the distribution of n traits in a single individual is discussed, and other problems are introduced.