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 for calculating concordant and discordant pairs 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 a pair of observations – (X_{i}, Y_{i}) and (X_{j}, Y_{j}):

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

For example, suppose a group of friends are playing darts. They identify their skill level as beginner, intermediate, or expert, and collect data on their accuracy - low, medium, and high.

Skill | Low | Med | High |
---|---|---|---|

Beginner | 10 | 2 | 1 |

Intermediate | 3 | 5 | 5 |

Expert | 3 | 7 | 3 |

A pair of observations is concordant if the subject who is higher on one variable is also higher on the other variable. A pair of observations is discordant if the subject who is higher on one variable is lower on the other variable.

Number of concordant pairs:

- Beginner, Low * Intermediate, Med = 50
- Beginner, Low * Expert, Med = 70
- Beginner, Low * Intermediate, High = 50
- Beginner, Low * Expert, High= 30
- Beginner, Med * Intermediate, High = 10
- Beginner, Med * Expert, High = 6
- Intermediate, Low * Expert, Med = 21
- Intermediate, Low * Expert, High = 9
- Intermediate, Med * Expert, High = 15

Number of discordant pairs:

- Intermediate, Low * Beginner, Med = 6
- Intermediate, Low * Beginner, High = 3
- Expert, Low * Intermediate, Med = 15
- Expert, Low * Intermediate, High = 15
- Expert, Low * Beginner, Med = 6
- Expert, Low * Beginner, High = 3
- Expert, Med * Beginner, High = 7
- Expert, Med * Intermediate, High = 35
- Intermediate, Med * Beginner, High = 5