Bayes' theorem of inverse probability is made the basis of a general equation for scoring objective examinations. The equation so obtained is evaluated by assuming a binomial distribution of examinee knowledge and guessing tendency. A graphical illustration of the application of this equation to a hypothetical test situation is presented. The limitations inherent in the use of Bayes' theorem make it inadvisable to recommend the practical use of the equation unless future experimental evidence indicates an increase in scoring validity which more than compensates for the increase in scoring difficulty.

Two new methods of item analysis are described. One involves the computation of the ϕ coefficient (correlation of a fourfold point distribution) and the other involves chi square. The only data required are the proportions of passing individuals in the upper and lower criterion groups, for the determination of ϕ, and in addition, N, for the determination of chi square. Abacs are presented for graphic solution of the two indices of validity, and tests of significance are provided.

This paper attempts to show in a non-mathematical way the influence of grouping on standard deviations and correlations, and advances empirical evidence to illustrate with what accuracy values corrected for grouping by Sheppard's correction approximate to values obtained from ungrouped data when the distributions are continuous. This enquiry gained its initial stimulus from the observation that many standard deviations and correlations reported by students of psychology and education are uncorrected for grouping and that frequently errors attributable to the grouping of data are not small when compared with errors of sampling.

The general problem of estimating correlated or uncorrelated factors is treated. It is specifically indicated wherein the prediction of oblique factors differs from that of orthogonal factors. A shortened method of estimation of correlated factors is developed.

Mental test data of Chrysostomm and of Garrett, Bryan, and Perl are reinvestigated in order to determine the shifts in test clusters with chronological age. It is found that the factors under survey tend to become more independent with increasing age.

In continuation of previous studies, the interreaction of two social classes is investigated from a somewhat different point of view. An exchange of results of activities of the two classes is considered, and a mathematical approach is outlined based on the use of such psychological concepts as the satisfaction function.

Fisher's method of maximum likelihood is applied to the problem of estimation in factor analysis, as initiated by Lawley, and found to lead to a generalization of the Eckart matrix approximation problem. The solution of this in a special case is applied to show how test fallability enters into factor determination, it being noted that the method of communalities underestimates the number of factors.

A method for computing the biserial correlation coefficients with the aid of punch card equipment is oultined. A numerical example and a work sheet layout is included in the presentation.

Two batteries of music tests were factored by the centroid method. From each battery three oblique factors were extracted and in each case were tentatively identified as tonal sensitivity, retentivity (memory for elements), and memory for form. The correlations of the music tests of one battery with subtests of Cattell's intelligence test and with tests of a literary nature are also reported.