The introduction of psychological tests for personnel selection in the British forces has given rise to several novel problems in statistical procedure. The solutions proposed are in the main extensions of devices already familiar in educational psychology. The more important are: (i) where the criterion yields a threefold classification only, a method of triserial correlation or of biserial correlation assuming point-distributions for the extremes; (ii) where the data on which validation has to be based are drawn from a selected sample, a simplified form of Pearson's equations to correct for selection; (iii) where the best line of demarcation has to be deduced from theoretical rather than practical considerations, a formula based on the principle of minimal discrepancy.

A dilemma was created for factor analysts by Ferguson (Psychometrika, 1941, 6, 323–329) when he demonstrated that test items or sub-tests of varying difficulty will yield a correlation matrix of rank greater than 1, even though the material from which the items or sub-tests are drawn is homogeneous, although homogeneity of such material had been defined operationally by factor analysts as having a correlation matrix of rank 1. This dilemma has been resolved as a case of ambiguity, which lay in (1) failure to specify whether homogeneity was to apply to content, difficulty, or both, and (2) failure to state explicitly the kind of correlation to be used in obtaining the matrix. It is demonstrated that (1) if the material is homogeneous in both respects, the type of coefficient is immaterial, but (2) if content is homogeneous but difficulty is not, the homogeneity of the content can be demonstrated only by using the tetra chorie correlation coefficient in deriving the matrix; and that the use of the phi-coefficient (Pearsonian r) will disclose only the nonhomogeneity of the difficulty and lead to a series of constant error factors as contrasted with content factors. Since varying difficulty of items (and possibly of sub-tests) is desirable as well as practically unavoidable, it is recommended that all factor analysis problems be carried out with tetrachoric correlations. While no one would want to obtain the constant error factors by factor analysis (difficulty being more easily obtained by counting passes), their importance for test construction is pointed out.

The predictive efficiency of a test used to select personnel is defined in terms of total effectiveness of the group thus selected, as compared with chance selection. The formula developed requires the use of an estimate of the ratio of average effectiveness of men selected to the average effectiveness of men not selected by the test. The predictive efficiency of the test varies directly with the magnitude of this ratio and also directly with the percentage rejected.

Mathematical expressions are derived for such concepts as stimulation of student by instructor, student-instructor rapport, and driving power of instructor, in terms of the student's and the instructor's foci of attention, their strength of concentration, and the intensity of the presentation and of the reception of details of subject matter. Under the assumption of normal distribution, the mathematical methods of combination and integration yield conclusions on summary integral effects of interrelations within the educational team. The psychological interpretation of the mathematical results thus obtained conforms with common sense. The main emphasis of the article is the exposition of how the mathematical method of combination and integration can be used to estimate the resultant effect of various independent combined simple factors acting independently within the individuals forming the educational team. No claim is made as to the absolute truthfulness and reliability of the psychological postulates used at the beginning stage of the mathematical analysis.

A simple method for extracting correlated factors simultaneously is described. The method is based on the idea that the centroid pattern coefficients for the sections of unit rank of the complete matrix may be interpreted as structure values for the entire matrix. Only the routine centroid average process is required.

A method whereby biographical or other questionnaire data of a purely qualitative nature may be used to predict success or failure on an independent criterion is presented. The method is not new but the present least-squares derivation and the transformation equation for punched card coding were not available in the literature. The proper weights are found to be proportional to the per cent of passers in the various categories. The method is suggested as a suitable substitute for non-linear approaches in connection with purely quantitative data as well. The implications of reweighting in connection with multiple regression is discussed. The lavish use of degrees of freedom makes cross-validation extremely desirable.

The choosing of a set of factors likely to correspond to the real psychological unitary traits in a situation usually reduces to finding a satisfactory rotation in a Thurstone centroid analysis. Seven principles, three of which are new, are described whereby rotation may be determined and/or judged. It is argued that the most fundamental is the principle of “parallel proportional profiles” or “simultaneous simple structure.” A mathematical proof of the uniqueness of determination by this means is attempted and equations are suggested for discovering the unique position.