Averaging Correlations: Expected Values and Bias in Combined Pearson rs and Fisher's z Transformations

R. A. Fisher's z (z'; 1958) essentially normalizes the sampling distribution of Pearson r and can thus be used to obtain an average correlation that is less affected by sampling distribution skew, suggesting a less biased statistic. Analytical formulae, however, indicate less expected bias in average r than in average z' back-converted to average r z' . In large part because of this fact, J. E. Hunter and F. L. Schmidt (1990) have argued that average r is preferable to average r z' . In the present study, bias in average r and average r z' was empirically examined. When correlations from a matrix were averaged, the use of z' decreased bias. For independent correlations, contrary to analytical expectations, average r z' was also generally the less biased statistic. It is concluded that (a) average r z' is a less biased estimate of the population correlation than average r and (b) expected values formulae do not adequately predict bias in average r z' when a small number of correlations are averaged.