Same Order, Different Scale

Pearson's $r$ measures how closely points follow a straight line. Sometimes that is the wrong question. If every extra hour of practice raises a score, but by less each time, the relationship is perfectly "more of one means more of the other", yet the points bend away from any straight line and Pearson's $r$ comes out below 1. And a single extreme value can drag Pearson's $r$ a long way.

Spearman's rank correlation keeps only the order of the values and throws away their sizes:

Replace each value in x by its rank within x: the smallest value gets rank 1, the next smallest rank 2, and so on. Rank y the same way, separately.
Compute Pearson's $r$ on the two lists of ranks, with the usual formula

r = \frac{\sum (a_i - \bar{a})(b_i - \bar{b})}{\sqrt{\sum (a_i - \bar{a})^2 \sum (b_i - \bar{b})^2}}

where $a_i$ and $b_i$ are now the ranks of $x_i$ and $y_i$ .

Ties. Equal values share the average of the ranks they would occupy. In [40, 10, 40, 25] the two 40s would take ranks 3 and 4, so each gets 3.5, and the ranks are [3.5, 1, 3.5, 2].

Task: write spearman(x, y) that returns Spearman's rank correlation rounded to 4 decimal places.

x and y have the same length $n \ge 2$ , and position $i$ in both lists describes the same item.
Each list contains at least two different values.

Same Order, Different Scale

Examples

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