Tests of Significance of Correlation Coefficients in the Absence of Bivariate Normal Populations

A computer program randomly sampled ordered pairs of scores from known populations that departed from bivariate normal form and calculated correlation coefficients from sample values. Even for small samples (N = 4, N = 10, N = 18), the t test of the hypothesis that the population correlation is zero was remarkably robust and reproduced the probabilities given by the sampling distribution of the t statistic with a high degree of accuracy for populations deviating extensively from bivariate normal form. Significance tests based on the Fisher r to Z transformation, in which hypotheses about non-zero values of the population correlation are tested, were less accurate, although in many cases probabilities were fairly close to the ones based on the normal distribution for appropriate significance levels. There is an interaction between the degree of departure of the population distribution from normality and the degree of departure of the population correlation from zero. Significance tests based on the r to Z transformation were fairly accurate despite extensive departures from normality when population correlations were close to zero. They were not accurate when population correlations deviated markedly from zero (.45 or higher).