The “pooled-within classes” coefficient of correlation (r_w) in partially segregated educational systems

The purpose of this paper is to point out the logical, theoretical and statistical problems associated with the computation of the “pooled-within” correlation coefficient in partially segregated educational systems. It is argued that, since a between-individual-within-classes relationship of Y to X cannot possibly exist in classes which are entirely homogeneous with respect to either X or Y, the question concerning this relationship should be addressed only to the subsample of heterogeneous classes. Consequently, the measurement of this relationship should be restricted to this subsample, thus yielding a new coefficient, r_wH, which is different from the usual coefficient, both conceptually and numerically. The remainder of the paper deals with the theoretical and statistical problems associated with the computation of r_wH instead of r_w, its interpretation as a sample statistic and its functional relationship to r_B and r_T, the “between groups” and “total” correlation coefficients, respectively.