A convenient, programable procedure is described that leads simultaneously to useful estimates of communality and of the number of factors. On the basis of actual data for two samples, properties of the communalities obtained by this procedure are compared with those obtained from a conventional principal axis analysis using diagonal entries of unity. The clinical scales of the MMPI, plus Si, are judged to provide reliable measurement on at least eight orthogonal dimensions, each well-marked by a different scale. Stable differences exist in the item-reliabilities of the ten scales studied. Neither of these consistencies in the data for the two samples was available from application of the conventional analysis. Tucker's procedure for examining factor congruence is found to be capable of yielding misleading results in the context of poor communality estimates.