Correlation Coefficients: Mean Bias and Confidence Interval Distortions

  • Richard L. Gorsuch (Fuller Theological Seminary)
  • Curtis S. Lehmann (Fuller Theological Seminary)


Non-zero correlation coefficients have non-normal distributions, affecting both means and standard deviations. Previous research suggests that z transformation may effectively correct mean bias for N's less than 30. In this study, simulations with small (20 and 30) and large (50 and 100) N's found that mean bias adjustments for larger N's are seldom needed. However, z transformations improved confidence intervals even for N = 100. The improvement was not in the estimated standard errors so much as in the asymmetrical CI's estimates based upon the z transformation. The resulting observed probabilities were generally accurate to within 1 point in the first non-zero digit. These issues are an order of magnitude less important for accuracy than design issues influencing the accuracy of the results, such as reliability, restriction of range, and N.

Keywords: confidence intervals, correlation coefficient, Fisher’s z transformation, Monte Carlo study, mean bias in correlation coefficients

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Published on
05 May 2011
Peer Reviewed