SL 4.10—Spearman's rank correlation

Spearman's rank correlation converts paired observations to ranks and measures whether high ranks tend to accompany high or low ranks. With no ties, r_s=1−6Σd²/[n(n²−1)], where d is the rank difference.

Rank ties need an agreed average-rank procedure, and a significance decision depends on the sample size and test rule. The coefficient describes a monotonic pattern, not necessarily a straight-line relationship.

If study hours and score have r_s=0.80, students with higher hours generally rank higher in score. A third variable, restricted range or a few influential pairs could still explain or distort the pattern.

Ranking removes units but not bias. A high r_s is not proof of causation, and a low value can hide a curved relationship that a rank coefficient does not capture.