What Is Spearman's Rank Correlation?

Spearman's rank correlation coefficient (ρ or rho) measures how well the relationship between two variables can be described by a monotonic function — one that consistently increases or decreases. It is computed by ranking each variable separately and then correlating the ranks, which makes it immune to outliers and non-linear but monotonic patterns that would defeat Pearson's linear correlation.

How the Calculation Works

Each variable is ranked from smallest (1) to largest, with ties receiving the average rank. Pearson's correlation formula is then applied to the two rank columns. When there are no ties, this simplifies to the familiar shortcut ρ = 1 − 6Σd² / (n(n² − 1)), where d is the difference between each pair's two ranks. The tie-aware rank-based computation used here is the general and exact version.

Interpreting ρ and Its Significance

ρ ranges from −1 to +1. A value near +1 means the ranks move together; near −1 they move in opposite directions; near 0 there is no monotonic relationship. The significance test converts ρ into a t-statistic with n − 2 degrees of freedom and reports a two-tailed p-value, telling you whether the observed association could plausibly be a random pattern for your sample size.