Spearman's Rank Correlation Coefficient

When the relationship between two variables is monotonic but not necessarily linear, Pearson's $r$ can understate the association. The Spearman rank correlation coefficient $r_s$ fixes this by measuring correlation between the ranks of the observations rather than the raw values.

To compute $r_s$ when no two $x$-values are tied and no two $y$-values are tied:

1. Rank the $x$-values from smallest (rank $1$) to largest (rank $n$). Do the same independently for the $y$-values.
2. For each pair $(x_i, y_i),$ compute the rank difference $d_i = R(x_i) - R(y_i).$
3. Apply the formula

$$r_s = 1 - \dfrac{6\sum\limits_{i=1}^n d_i^2}{n(n^2-1)}.$$

The value of $r_s$ always lies between $-1$ and $1.$ It equals $+1$ exactly when the two rankings are identical (perfect monotonic increase) and $-1$ when one ranking is the exact reverse of the other (perfect monotonic decrease).

Tiny example. With $n=3$ and the data

$$x: 1,\;2,\;3 \quad\longrightarrow\quad \text{ranks } 1,2,3,$$ $$y: 10,\;30,\;20 \quad\longrightarrow\quad \text{ranks } 1,3,2,$$

the rank differences are $d_i = 0,\,-1,\,1,$ so $\sum d_i^2 = 0+1+1 = 2$ and

$$r_s = 1 - \dfrac{6 \cdot 2}{3(3^2-1)} = 1 - \dfrac{12}{24} = 0.5.$$