Spearman's rank correlation coefficientEdexcel International A Level Further Maths: Revision notes

Spearman's rank correlation coefficient, $r_s$, measures how well the ranks of one variable agree with the ranks of another. It takes values from $-1$ to $+1$. $r_s=+1$: the two rankings are identical. $r_s=-1$: one ranking is the exact reverse of the other. $r_s$ near 0: no consistent agreement. Use it when the data are already ranks (for example two judges), or when the relationship is monotonic (always increasing or always decreasing) but not necessarily linear, or when the data are not from a bivariate Normal distribution.

Rank each variable separately, using the same direction for both (rank 1 for the largest in both, or rank 1 for the smallest in both). The direction can be chosen freely, as long as it is the same for both variables. Ties: if two or more values are equal, give each the mean of the ranks they would have occupied. For example, values in 4th and 5th place that are equal each get rank $4.5$, and the next value is ranked 6th. Questions with numerical ties will not be set, but you should know this rule and be able to state the ranks.

Find the difference $d=\text{rank}_x-\text{rank}_y$ for each pair and then $\sum d^2$: $r_s=1-\frac{6\sum d^2}{n(n^2-1)},$ where $n$ is the number of pairs. Worked example: six dancers are ranked by two judges with $d=-2,1,1,-1,-1,0$. $\sum d^2=8$ and $r_s=1-\frac{6\times8}{6\times35}=1-\frac{48}{210}=0.771$. Check: $\sum d$ must be 0, and $\sum d^2$ is always an even number when there are no ties.

Always interpret in context. $r_s$ close to $+1$: strong positive correlation between the ranks, so items ranked high on one tend to be ranked high on the other. Close to $-1$: strong negative correlation, so high on one means low on the other. Close to 0: little or no agreement. Reversing one of the rankings changes the sign of $r_s$ but not its size. Correlation is not causation: a strong $r_s$ does not show that one variable causes the other, as a third variable may affect both. Size of $n$ matters too: with few pairs a large $r_s$ can occur by chance (a test for this is the next step in the unit).

It uses only the order of the values, so it ignores how far apart they are and loses information. A very large change in one value can have the same effect as a small one. It measures only the strength of a monotonic relationship. A relationship that rises then falls can give $r_s$ near 0 even though the variables are strongly related. It does not show cause. With small $n$ the value is unreliable. Advantage over the product moment correlation coefficient: it is less affected by outliers and needs no assumption of a bivariate Normal distribution.