Many research studies involving Pearson correlations are conducted in settings where one of the two variables has a restricted range in the sample. For example, this situation occurs when tests are used for selecting candidates for employment or university admission. Often after selection, there is interest in correlating the selection variable, which has a restricted range, to a criterion variable. The focus of this research was to compare Alexander, Alliger, and Hanges’s (1984) formula to Thorndike’s (1947) formula and population values using Monte Carlo simulation when the assumption of normal distribution is violated in a particular way.

In both Thorndike’s and Alexander et al.’s correction formulas, values for the variances in the restricted and the unrestricted situations are required. For both formulas, the variance in restricted situations was a sample estimate. In the Monte Carlo simulation, the difference between the two approaches was that in Thorndike’s formula, the variance in the unrestricted situation was the population variance from the exogenous variable, whereas in Alexander et al.’s approach, the population variance was estimated based on the sample variance in the restricted situation. In the simulation, robustness situations were created from non-normal distributions for predicted group membership in a classification problem.

As expected, Thorndike’s corrected correlation values were more accurate than Alexander et al.’s corrected correlation values, and Thorndike’s formula had a smaller standard error of estimates. Absolute values of the mean differences between the estimated and population correlations for Alexander et al.’s approach compared to Thorndike’s approach in robustness situations ranged from 1.37 to 2.15 larger. Nevertheless, Alexander et al.’s approach, which is based only on estimated variances, appears to be a worthwhile correction in most of the simulated situations with a few notable exceptions for non-normal distributions.