What is Spearman Correlation 🌱

Suppose we have two random variables $X$ and $Y$ with $n$ paired samples from their joint distribution. We can define the ranking, $\mathrm{rg}{X}$,ofthevariable$X$for$X_1,…,X_n$tobethepositionofeach$X_i$when$X_1,…,X_n$areorderedfromleasttogreatest.Forexample,supposewehave$(X_1,X_2,X_3,X_4) = (7,9.3,2,-1)$,thentheleasttogreatestorderingof$X$is$(-1,2,7,9.3)$and$\mathrm{rg}{X} = (4, 3, 1, 2)$.

The Spearman correlation $r_s$ is then the same thing as the Pearson correlation coefficient between the rank variables: $r_s={\rho }_{{rg}_X,{rg}_Y}=\frac{\mathrm{cov}({rg}_X,{rg}_Y)}{{\sigma }_{{rg}_X}{\sigma }_{{rg}_Y}}$