This paper is concerned with a central concept in social measurement such as opinion polls, the measurement of attitudes, the prediction of political elections, the measurement of moral values, the measurement of consumer preferences, the measurement of utility, and the measurement of aesthetic values. The concept is that of the discriminal dispersion and its interesting effects in the prediction of choice.

Three sources of variation in experimental results for a test are distinguished: trials, persons, and items. Unreliability is defined only in terms of variation over trials. This definition leads to a more complete analysis than does the conventional one; Spearman's contention is verified that the conventional approach—which was formulated by Yule—introduces unnecessary hypotheses. It is emphasized that at least two trials are necessary to estimate the reliability coefficient. This paper is devoted largely to developing lower bounds to the reliability coefficient that can be computed from but a single trial; these avoid the experimental difficulties of making two independent trials. Six different lower bounds are established, appropriate for different situations. Some of the bounds are easier to compute than are conventional formulas, and all the bounds assume less than do conventional formulas. The terminology used is that of psychological and sociological testing, but the discussion actually provides a general analysis of the reliability of the sum of n variables.

This paper describes a simple orthogonal multiple factor approximation procedure that involves no inversion of the sign of negative residuals, the estimation of only as many communalities as there are factors, and none or only a few minor rotations of the axes in an attempt to obtain a “meaningful” solution. It also suggests a technique for the estimation of those communalities that must be estimated. The factor loadings obtained by means of this procedure, which we shall designate as the pre-selection procedure, are affected by the order in which the factors are obtained, showing a reduction in variance accounted for by each successive factor, as is characteristic of the centroid, bi-factor, and principal factor solutions. The entire procedure takes considerably less time than that involved in the orthodox centroid method alone.