A short note on the maximal point-biserial correlation under non-normality

The aim of this paper is to derive the maximal point-biserial correlation under non-normality. Several widely used non-normal distributions are considered, namely the uniform distribution, t-distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point-biserial correlation, depending on the non-normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non-symmetric around p=.5, and may still lie in the range from -1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point-biserial correlation coefficients based on popular beliefs that the maximal point-biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.