Averaging Correlations: Expected Values and Bias in Combined Pearson rs and Fisher's z Transformations
The Journal of General Psychology, 125 (3):
245--261 (1998)DOI: 10.1080/00221309809595548
Abstract
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 rz' . In large part because of this fact, J. E. Hunter and F. L. Schmidt (1990) have argued that average r is preferable to average rz' . In the present study, bias in average r and average rz' was empirically examined. When correlations from a matrix were averaged, the use of z' decreased bias. For independent correlations, contrary to analytical expectations, average rz' was also generally the less biased statistic. It is concluded that (a) average rz' is a less biased estimate of the population correlation than average r and (b) expected values formulae do not adequately predict bias in average rz' when a small number of correlations are averaged.