Experimental studies of the successive changes (frequently represented by curves describing laws of learning and other similar functional relationships) in a criterion variable accompanying experimental variations in a given “treatment,” and experimental comparisons of such changes for different populations or for different treatments, constitute a large and important class of psychological experiments. In most such experiments, no attempt has been made to analyze or to make allowance for errors of sampling or of observation. In many others, the techniques of error analysis that have been employed have been inefficient, inexact, or inappropriate. This paper suggests tests, using the methods of analysis of variance, of certain hypotheses concerning trends and trend differences in sample means obtained in experiments of this general type. For means of successive independent samples, tests are provided of the hypotheses: (H1) that there is no trend, or that the trend is a horizontal straight line, (H3) that there is a linear trend, (H5) that the trend is as described by a line not derived from the observed means, and (H7) that the trend is as described by a line fitted to the observed means. Tests are also provided of similar hypotheses (H2, H4, H6, and H8, respectively) for means of successive measurements of the same sample. Finally, tests are provided of the null hypotheses that there is no difference in trend in two series of means: (H9) when each mean in each series is based on an independent sample, (H10) when each pair of corresponding means is based on an independent sample, (H11) when each series of means is based on an independent sample, and (H12) when both series are based on a single sample.

In certain types of experiment it frequently happens that we have a block of data consisting of several trials on the same individuals. Using the methods of estimation provided by the analysis of variance, estimates of reliability are derived for this case, and the conditions under which each is valid are discussed. Various relations between these estimates and the product-moment coefficient of correlation are obtained.

This paper shows the formulation of nine methods of estimating the unknown communalities. Each of these methods has been used on experimental data and the results tabulated for comparison. The results show that the most accurate approximations are obtained from the Centroid No . 1 and the Graphical methods.

Several of the more popular statistics texts used in psychology are reviewed with respect to their treatment of the measurement and interpretation of skewness. Some areas in psychology where measures of skewness of distributions may yield significant information are indicated. Tables of 1%, 2% and 5% points of moment measures of skewness and tables of the 1% and 5% points of the difference in skewness of two uncorrelated distributions are presented. These tables are approximations based on the approximate normality of skewness in large samples from normal populations. The limited confidence with which these tables can be used in the absence of exact knowledge of the distribution function of the underlying statistics is indicated.

In this modification of the Aitken Pivotal Condensation Method for obtaining partial regression weights and multiple correlation coefficients, the unit matrices used by Aitken are eliminated, with a resultant decrease in the computational work. Through a simplified arrangement of successive matrices, each one is reduced by the same simple rule. Other modifications also contribute to ease of computation and facilitate checking.

The nervous system, and particularly the cerebral cortex, is examined in certain of its activities, it being treated as a purely physical dynamic system. It is shown mathematically that if, as seems likely, the cerebral cortex can undergo manifold changes of neuronic organization, then it follows that these changes must automatically lead to the development of more and more states of equilibrium, the process being unavoidable and largely irreversible. The theorem may be established on basic physical principles without appeal to special physiological details. The possible significance of this observation is indicated.

In the absence of sound units of measurement, the data obtained by use of a psychometric instrument may sometimes be so irregular that they cannot be accepted as a good representation of the true function of the trait measured. Frequently, ordered categorical or ranked data and other irregular distributions are obtained to represent traits that are obviously continuous and probably unskewed functions. In such cases, the data may be improved in the direction of a better approximation of the true function by converting them to normal distribution form. A table of mean deviates of various portions of the unit normal curve is presented to facilitate this conversion.

Two formulas are presented for judging the significance of the difference between correlated proportions. The chi square equivalent of one of the developed formulas is pointed out.