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.
Headrick, Todd C. "A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation." Open Journal of Statistics 6 (Nov 2016): 1025-1027. doi:10.4236/ojs.2016.66082.