In applied settings, it is typical to find a correlation between a predictor and some criterion. Unfortunately, if the predictor is used to choose the subjects, the range of the predictor is seriously reduced. This restricts the observed correlation to be less than would be observed in the full range of the predictor. A correction for this problem is well known: Let R the unrestricted correlaton, r the restricted correlation, S the unrestricted standard deviation, s the restricted standard deviation, then
R = rS/(s sqrt(1-r^2 + r^2(S^2/s^2)).