Spearman's rank correlation

Spearman's rank correlation uses ranks (1st, 2nd, 3rd…) instead of the actual values. It is used:

• when a variable cannot be measured, only ranked: beauty, honesty, the quality of a song;
• when the data have extreme values, which would distort r;
• when the relationship is clear but not linear.

The formula:

rs = 1 – 6ΣD² ÷ (n³ – n), where D is the difference between the two ranks of each item, and n is the number of items.

The steps: rank each variable, subtract the ranks to get D, square each D, add the squares, and apply the formula. Like r, it lies between –1 and +1. For example, with 6 items and ΣD² = 10: rs = 1 – 60 ÷ (216 – 6) = 1 – 60 ÷ 210 = 0.71.

In the book's example, three judges rank the same contestants. Comparing each pair of judges' rankings shows which two agree most (the highest rs) and which disagree (a negative rs).

If two or more items have equal values (a tie), each gets the average of the ranks they would have taken. If two items would take ranks 4 and 5, both get 4.5. For each tied group of m items, a correction of (m³ – m) ÷ 12 is added to ΣD².

Rank correlation uses less information than r (only the order, not the actual sizes). So for data that are precisely measured, Pearson's r is more accurate.