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.

Spearman's rho 是实践中最常用的依赖度量之一，用于描述两个随机变量之间的关联。然而，在至少一个随机变量是离散的情况下，Spearman 的相关性通常有界并限制在\([-1,1]\)的子区间内。因此，当接近可达到的最高值时，Spearman 的 rho 的小正值实际上可能支持强烈的正相关性。同样，Spearman 的 rho 的轻微负值实际上可能意味着强烈的负相关。在本文中，当至少一个随机变量是离散的时，我们推导出了 Spearman's rho 的最可能的上限和下限。我们在一些实际相关的情况下说明了获得的下限和上限。