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

• Published in: The Journal of general psychology Vol. 125; no. 3; pp. 245 - 261
• Main Authors: Corey, David M., Dunlap, William P., Burke, Michael J.
• Format: Journal Article
• Language: English
• Published: Provincetown, Mass., etc Taylor & Francis Group 01.07.1998; Journal Press, etc; Taylor & Francis Inc
• Subjects: Averaging; Bias; Estimates; Estimation bias; Expected values; Experiments; Meta-analysis; Population; Psychology; Reservoirs; Sample size; Social sciences
• DOI: 10.1080/00221309809595548
• ISSN: 0022-1309

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.