Use concordant and discordant pairs to describe the relationship between pairs of observations. To calculate the concordant and discordant pairs, the data are treated as ordinal, so ordinal data should be appropriate for your application. The number of concordant and discordant pairs are used in calculations for Kendall's tau, which measures the association between two ordinal variables.
The procedure compares the classifications for two variables (for example, X and Y) on the same two items. If the direction of classifications is the same, the pairs are concordant. For example, both X and Y rate item 1 higher than item 2. If the direction of the classification is not the same, the pair is discordant. For example, X rates item 1 higher than item 2 but Y rates item 1 lower than item 2.
For example, judges at a singing competition rate 5 contestants on a scale from 1 to 5, where 1 is the best.
Arranging the ratings in order helps to determine which pairs are concordant and which are discordant.
Compare the ratings for Juan and Sophia to determine if this pair of the judges' ratings are concordant or discordant. Judge X's rating for Juan (1) is less than their rating for Sophia (2); Judge Y's rating for Juan (2) is less than their rating for Sophia (3). Because both judges' ratings for Juan are less than their ratings for Sophia, this pair of ratings is concordant. The following shows the comparison of all pairs of ratings that include Juan:
7 of the pairs are concordant. 3 of the pairs are discordant.