Spearman Rank Correlation Coefficient

HISTORY

Spearman, Charles was a psychologist. In 1904 he introduced for the first time the rank correlation coefficient. Often called the ρ of Spearman, it is one of the oldest rank statistic.

MATHEMATICAL ASPECTS

Let ${ (X_1,X_2,\ldots,X_n) }$ and ${ (Y_1,Y_2,\ldots,Y_n) }$ be two samples of size n. ${ R_{X_i} }$ denotes the rank of X _i compare to the other values of the X sample, for ${ i=1,2,\ldots,n }$. ${ R_{X_i}=1 }$ if X _i is the smallest value of X, ${ R_{X_i}=2 }$ if X _i is the second smallest value, etc., until ${ R_{X_i}=n }$ if X _i is the largest value of X. In the same way, ${ R_{Y_i} }$ denotes the rank of Y _i , for ${ i=1,2,\ldots,n }$.

The Spearman rank correlation coefficient, generally denoted by ρ, is defined by:

$$ \rho...