The error function of the theoretical learning curve developed by Thurstone has been fitted to maze data from rats that have been subjected to cerebral cortical insult. There is a high degree of relationship between the number of errors necessary to complete learning as predicted from the curve and the experimental measure of learning, total-errors-minus-errors-first-trial. The procedure provides a method of analysis for individual learning records and it may be used to shorten the training period. The goodness of fit indicates that the underlying logic of the learning curve is plausible.

The problem of fitting a least-squares linear subspace to a higher dimensional set of points is solved by the use of matrix approximation results, both when the subspace is required to pass through a given point, and when it is not so restricted. In the latter case the subspace is shown to pass through the centroid of the point set. It is noted that the principal axes of the set of points enter into the problem in a natural manner.

The basic conditions and characteristics of measurement are outlined. The concepts entering into the measurement must be operationally defined; there must be a satisfactory representation of the character to be measured, and a quantitative comparison between the observed degree of the character and some reference point or scale. A conception of measurement which is too narrow would unnecessarily restrict investigation in the fields of psychology and education.

This paper presents three theorems concerning the relation between results with obtained and corrected correlation coefficients in the Thurstone method of multiple factor analysis. (1) The rank of the correlational matrix, and thus the number of factors involved, is unaffected by correcting the obtained coefficients for attenuation. (2) The communality of a variable when the obtained coefficients have been corrected for attenuation is equal to the communality with obtained coefficients divided by the reliability coefficient of the variable. (3) The relationship is shown between the factorial matrix of a correlational matrix of raw correlation coefficients and the factorial matrix of a correlational matrix of corrected correlation coefficients, and a method of obtaining either of these factorial matrices from the other is indicated.

The Bi-factor Method of factor analysis is described and illustrated with a small group of fourteen tests. A detailed illustration is given of how the method may be modified to the case of overlapping group factors. It is advocated that the Bi-factor pattern in unmodified form be used to determine the adequacy of tests for the measurement of unitary traits.

A seven-decimal table is presented of the area under the unit normal curve, for abscissae expressed in terms of the “probable error” or PE. From the method of calculation, the partial verification by means of other tables, and the safeguards taken in the routine of computation, it is safe to consider the table reliable. Errors in previously published tables are pointed out.