A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation

Abstract

This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.