Bounds on Spearman’s rho when at least one random variable is discrete

Spearman’s rho is one of the most popular dependence measures used in practice to describe the association between two random variables. However, in case of at least one random variable being discrete, Spearman’s correlations are often bounded and restricted to a sub-interval of [- 1 , 1]. Hence, small positive values of Spearman’s rho may actually support a strong positive dependence when getting close to its highest attainable value. Similarly, slight negative values of Spearman’s rho can actually mean a strong negative dependence. In this paper, we derive the best-possible upper and lower bounds for Spearman’s rho when at least one random variable is discrete. We illustrate the obtained lower and upper bounds in some situations of practical relevance.