Concordant pair

In statistics, a concordant pair is a pair of observations, each on two variables, (X₁,Y₁) and (X₂,Y₂), having the property that

${\displaystyle \operatorname {sgn}(X_{2}-X_{1})\ =\operatorname {sgn}(Y_{2}-Y_{1}),\ }$

where "sgn" refers to whether a number is positive, zero, or negative (its sign). Specifically, the sign function, often represented as sgn, is defined as:

${\displaystyle \operatorname {sgn} x=\left\{{\begin{matrix}-1&:&x<0\\0&:&x=0\\1&:&x>0\end{matrix}}\right.}$

That is, in a concordant pair, both elements of one pair are either greater than, equal to, or less than the corresponding elements of the other pair.

In contrast, a discordant pair is a pair of two-variable observations such that

${\displaystyle \operatorname {sgn}(X_{2}-X_{1})\ =-\operatorname {sgn}(Y_{2}-Y_{1}).\ }$

That is, if one pair contains a higher value of X then the other pair contains a higher value of Y.