What Is Spearman's Rank Correlation Coefficient and How Do You Calculate It?

Spearman's rank correlation coefficient, denoted by $r_s$, measures the strength and direction of a monotonic relationship between two variables using their ranks rather than their original values. It ranges from $-1$ to $1$.

How Spearman's Rank Works

A relationship is monotonic when one variable generally increases as the other increases, or generally decreases as the other increases. Unlike Pearson's $r$, Spearman's $r_s$ does not require the relationship to be linear and is less sensitive to outliers.

Because only order matters, any transformation that preserves the rankings leaves $r_s$ unchanged. This makes the method appropriate for ordinal data and curved but consistently increasing or decreasing patterns.

Calculating $r_s$

1. Rank both variables consistently, usually from smallest to largest.
2. For each pair, calculate the rank difference $d$.
3. Square each difference and find $\sum d^2$.
4. If there are no tied ranks, use

$$r_s=1-\frac{6\sum d^2}{n(n^2-1)},$$

where $n$ is the number of paired observations.

For example, suppose the two sets of ranks are $1,2,3,4,5$ and $1,3,2,5,4$. The squared differences are $0,1,1,1,1$, so :

$$r_s=1-\frac{6(4)}{5(5^2-1)}=0.8.$$

This indicates a positive monotonic association. The rankings mostly move together, but two adjacent pairs are reversed. The coefficient therefore remains positive and fairly strong rather than perfect.

If values are tied, assign each tied value the average rank it would occupy. Do not use the shortcut formula unchanged when tied ranks occur; instead, calculate the correlation of the averaged rank lists. In IB Mathematics: Applications and Interpretation, $r_s$ is normally found using technology.

Exam Technique

This is SL 4.10 and may be tested on either GDC-active SL paper. Show consistent rankings, average tied ranks, state $r_s$, and interpret its sign and strength in context. A common misconception is that correlation proves causation but it does not.