Spearman's rank correlation coefficient

Remarks

The closed formula holds because the ranks of each variable are the numbers $1, \dots, n$, whose mean and variance are known. For random variables $X$ and $Y$ with continuous distribution functions $F$ and $G$, the population version is the correlation of $F(X)$ and $G(Y)$, which equals $12 \, E[F(X) G(Y)] - 3$.

Example: $x = (1, 2, 3)$ and $y = (1, 3, 2)$. The ranks are $r = (1, 2, 3)$ and $s = (1, 3, 2)$, so $d = (0, -1, 1)$ and $\rho_s = 1 - 6 \cdot 2 / (3 \cdot 8) = 1/2$. For $y = (1, 4, 9)$ instead, $d = 0$ and $\rho_s = 1$.

Non-example: for data with ties, such as $y = (1, 1, 2)$, the ranks are not defined by the clause above. The usual fix gives tied values the average of their ranks and takes the Pearson correlation; the formula with $\sum d_i^2$ is then no longer exact.

The Kendall rank correlation coefficient is another rank correlation, based on pairs of observations.