A multiple-factor analysis was made of a battery of 42 tests of verbal abilities administered to 119 college adults. Where necessary, the distributions of test scores were normalized before the inter-test correlations were computed. Thurstone's M (Memory or Rote Learning) factor has been confirmed, but his V (Verbal Relations) factor seems to have been split into two or possibly three factors, C, J, and G; and his W (Word Fluency) factor has been split into two factors, A and E. The C factor seems to represent the richness of the individual's stock of linguistic responses, and the J factor seems to involve the ability to handle semantic relationships. No satisfactory interpretation can as yet be made of the G factor. The A factor seems to correspond to the speed of association for common words where there is a high degree of restriction as to appropriate responses. The E factor is described as an associational facility with verbal material where the only restriction is that the responses must be syntactically coherent. The new factors are: F, facility and fluency in oral speech; H, facility in attaching appropriate names or symbols to stimuli; and D, speed of articulatory movements.

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student “t” and the Fisher “z” tests to a wider range of problems.

A previous mathematical study of a situation, in which the behavior of a larger group of individuals is controlled by a smaller group, is generalized for the case when the “activity” of the individuals in the group is continuously graded. The existence of two possible social configurations and of sudden transitions from one configuration to another are found in this case also.

This paper discusses the influence of test difficulty on the correlation between test items and between tests. The greater the difference in difficulty between two test items or between two tests the smaller the maximum correlation between them. In general, the greater the number of degrees of difficulty among the items in a test or among the tests in a battery, the higher the rank of the matrix of intercorrelations; that is, differences in difficulty are represented in the factorial configuration as additional factors. The suggestion is made that if all tests included in a battery are roughly homogeneous with respect to difficulty existing hierarchies will be more clearly defined and meaningful psychological interpretation of factors more readily attained.

On viewing Thurstone's psychophysical scale from the point of view of the mathematical theory of one-parameter continuous groups, it is seen that a variety of different psychological or statistical assumptions can all be made to lead to a scale possessing similar properties, though requiring different computational techniques for their determination. The natural extension to multi-dimensional scaling is indicated.