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,. 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,. In the present study, bias in average r and average r, 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, when a small number of correlations are averaged.