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.

Specifically, there are two sets of paired observations – (X_{i}, Y_{i}) and (X_{j}, Y_{j}):

- The pairs are concordant if X
_{i}> X_{j}and Y_{i}> Y_{j}or X_{i}< X_{j}and Y_{i}< Y_{j} - The pairs are discordant if X
_{i}> X_{j}and Y_{I}< Y_{j}or X_{i}< X_{j}and Y_{i}> Y_{j}

For example, judges at a singing competition rate 5 contestants on a scale from 1 to 5, where 1 is the best.

Judge | Robert | Juan | Sophia | Helena | Marie |
---|---|---|---|---|---|

Judge X | 3 | 1 | 2 | 4 | 5 |

Judge Y | 1 | 2 | 3 | 5 | 4 |

Arranging the ratings in order helps to determine which pairs are concordant and which are discordant.

Judge | Juan | Sophia | Robert | Helena | Marie |
---|---|---|---|---|---|

Judge X | 1 | 2 | 3 | 4 | 5 |

Judge Y | 2 | 3 | 1 | 5 | 4 |

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:

- Juan vs Sophia: X
_{J}, X_{S}= 1, 2 and Y_{J}, Y_{S}= 2, 3; 1 < 2 and 2 < 3 so the pair is concordant. - Juan vs Robert: X
_{J}, X_{R}= 1, 3 and Y_{J}, Y_{R}= 2, 1; 1 < 3 and 2 > 1 so the pair is discordant. - Juan vs Helena: X
_{J}, X_{H}= 1, 4 and Y_{J}, Y_{H}= 2, 5; 1 < 4 and 2 < 5 so the pair is concordant. - Juan vs Marie : X
_{J}, X_{M}= 1, 5 and Y_{J}, Y_{M}= 2, 4; 1 < 5 and 2 < 4 so the pair is concordant.

Then, compare the remaining pairs:

- Sophia vs Robert: X
_{S}, X_{R}= 2, 3 and Y_{S}, Y_{R}= 3, 1; 2 < 3 and 3 > 1 so the pair is discordant. - Sophia vs Helena: X
_{S}, X_{H}= 2, 4 and Y_{S}, Y_{H}= 3, 5; 2 < 4 and 3 < 5 so the pair is concordant. - Sophia vs Marie: X
_{S}, X_{M}= 2, 5 and Y_{S}, Y_{M}= 3, 4; 2 < 5 and 3 < 4 so the pair is concordant. - Robert vs Helena: X
_{R}, X_{H}= 3, 4 and Y_{R}, Y_{H}= 1, 5; 3 < 4 and 1 < 5 so the pair is concordant. - Robert vs Marie: X
_{R}, X_{M}= 3, 5 and Y_{R}, Y_{M}= 1, 4; 3 < 5 and 1 < 4 so the pair is concordant. - Helena vs Marie: X
_{H}, X_{M}= 4, 5 and Y_{H}, Y_{M}= 5, 4; 4 < 5 and 5 > 4 so the pair is discordant.

7 of the pairs are concordant. 3 of the pairs are discordant.