It is shown that by making the assumption that the knowledge of the test-item and the knowledge of the entire test are both distributed normally, the correlation coefficient between any item and the entire test can be expressed as a function solely of two proportions — the percentage of a high-scoring group passing the item and the percentage of a low-scoring group passing the item. This function is expressed graphically as a family of curves for each of two conditions — where the high-scoring and low-scoring groups are samples of the highest and the lowest quarters respectively, and where they are samples from the upper and lower halves. It is shown, moreover, that two other common measures of item validity, the upper-lower difference and the critical ratio of the upper-lower difference, may be drawn on the same coordinate axes.

It is noted that the average inter-item correlation, which represents the internal consistency of a test, yields a unique estimate of test reliability. A close approximation to this average is given by a formula which requires the correlation of each item with the total score and the standard deviation of each item. The formula is especially useful in those instances where the number of items is small and where the variation in item sigmas should not be neglected.

In tetrachoric correlation there is a four-fold distribution based on the four combinations of A’s and not —A’s and B’s and not —B’s. The analogous problem in test construction is that of determining the relationship between two items or test questions, upon which all individuals are scored pass or fail.

The assumption of specific operation times leads to equations similar to those which are the basis for a factor analysis but in which the time scores replace item scores. An analysis of time scores is shown to lead to results equivalent, in a first approximation, to those obtained from an analysis of item scores. It is suggested that by modifying the factor technique it may be possible to use additional information to check with the results from an analysis.

A machine to expedite matrix multiplication has been developed by modifying the International Business Machines Corporation scoring machine. The principles and operation of the machine are described, and time and accuracy estimates are indicated.

The fundamental factor theorem is developed in matrix form for the case of correlated factors. The properties of the correlated factor system are discussed, and some effects of sampling error considered. The psychological meaning of correlated factors is discussed, and several mechanisms by which general factors may operate in the factorial system are indicated.

A battery of thirty-six tests was given to a group of high-school seniors. The factorial analysis reveals essentially the same primary factors that were found in previous studies. The test battery reveals a simple structure.

The problem in multiple correlation work of nonsense results attributable to linear dependence of variables, which has been discussed by Ragnar Frisch in relation to economic data, is presented from the standpoint of its significance in psychological research. It is shown that a symmetric correlation determinant with unity in the diagonal cells can vanish only when there is a first-order or partial correlation of unity between one pair of the variables. On the basis of this result, it is argued that the problem should be expected to cause less difficulty in the field of psychology than in economics and that psychologists should be able to avoid the pitfall by bringing to bear their knowledge of the variables with which they are working.

An n-rowed correlation matrix may be thought of as an ellipsoid in n-dimensional space with its center at the origin. The principal components of the matrix are essentially the semi-axes of the ellipsoid. A direct and simple method of computing the lengths and directions of these semi-axes is presented.

An attempt is made to define and treat analytically the concept of individual freedom in a society. Two possible definitions are briefly discussed. One takes as a measure of freedom the ratio (w0 − w)w0, where w0 is the maximum amount of work that a person can physically perform per unit time and w is the amount of work which he has actually to perform per unit time in a given society. The other definition takes as a measure of freedom that fraction of an individual's time during which he can indulge in any activity of his own choice without interfering with other individuals. Expressions are derived by way of illustration, giving the individual freedom in terms of other parameters which characterize the social structure.

The Kuder-Richardson reliability coefficient is derived in a manner independent of that originally given. Various alternative forms applicable to special situations are exhibited with the purpose of making them available to others interested in using this formula. A simplification in computation is suggested for use with a calculating machine.