Concordant pair

For paired ordered observations $(x_i,y_i)$ and $(x_j,y_j)$, the pair is concordant when $(x_i-x_j)(y_i-y_j)>0$: both coordinates increase in the same ordering. It is a discordant pair when this product is negative. Equality gives a tie in at least one coordinate. Counting these cases compares orderings without assuming meaningful equal distances between categories.