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Let $$\rho_S = \frac{\sum_{i=1}^{n}(R_i-\overline R)\cdot(S_i-\overline S)}{\sqrt{\sum_{i=1}^{n}(R_i-\overline R)^2\cdot \sum_{i=1}^{n}(S_i-\overline S)^2}}$$

There $R_i$ rank of $X_i$ and $S_i$ rank of $Y_i$. There $X_i$, $Y_i$ two samples.

Sort pairs $(R_i,S_i)$ by first coordinate and get pairs $(i, T_i)$

How to prove that $$\rho_S=1-\frac{12}{n^3-n}\sum_{i<j}(j-i)\cdot I(T_i>T_j)$$

Николай Алексеевич Терехов
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  • Using the characterization of covariance I give at https://stats.stackexchange.com/a/18200/919 (and prove at https://stats.stackexchange.com/a/504234/919), the algebra becomes relatively easy, because you don't have to deal with the means $\bar R$ and $\bar S.$ – whuber Jan 15 '21 at 14:14

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