Definition:Kendall's Coefficient of Concordance
Definition
Kendall's coefficient of concordance is a test for consistency of more than $2$ sets of rankings.
Let $m$ judges independently award ranks $1$ to $n$ to a set of $n$ competitors.
Let $s_i$ be the sum of the rankings awarded to competitor $i$.
The mean $M$ of the values of $s_i$ is $\dfrac 1 2 m \paren {n + 1}$.
The sum of the squares of the deviations from $M$ is given by:
- $S = \ds \sum_{i \mathop = 1}^n \paren {s_i - M}^2$
and Kendall's coefficient of concordance is given by:
- $W = \dfrac {12 S} {m^2 n \paren {n^2 - 1} }$
Kendall's Coefficient of Concordance is between $0$ and $1$
Kendall's coefficient of concordance $W$ fulfils the following inequality:
- $0 \le W \le 1$
where:
- $0$ indicates complete disagreement
- $1$ indicates complete agreement.
Examples
Arbitrary Example 1
Consider the $3$ competitors $\text {Xavier}$, $\text {Yuri}$ and $\text {Zal}$, who are demonstrating their skills to the $4$ judges $\text {Araminta}$, $\text {Boecluvius}$, $\text {Coriolanius}$ and $\text {Derek}$.
The following table shows the ranking afforded the competitors by each judge, with the row summations per competitor:
| $\textit {Judge}$ | ||
| $\textit {Competitor}$ | $\begin{array} {r {{|}} cccc {{|}} c}
& \text A & \text B & \text C & \text D & s_i \\ \hline \text X & 1 & 1 & 2 & 3 & 7 \\ \text Y & 3 & 2 & 3 & 1 & 9 \\ \text Z & 2 & 3 & 1 & 2 & 8 \\ \hline \end{array}$ |
Kendall's coefficient of concordance is $\dfrac 1 {16}$.
Arbitrary Example 2
Consider the $4$ competitors $\text {Wilhelmina}$, $\text {Xanthippe}$, $\text {Yondalla}$ and $\text {Zena}$, who are demonstrating their skills to the $4$ judges $\text {Ariadne}$, $\text {Boudicca}$, $\text {Constantine}$ and $\text {Donald}$.
The following table shows the ranking afforded the competitors by each judge, with the row summations per competitor:
| $\textit {Judge}$ | ||
| $\textit {Competitor}$ | $\begin{array} {r {{|}} cccc {{|}} c}
& \text A & \text B & \text C & \text D & s_i \\ \hline \text X & 1 & 1 & 4 & 4 & 10 \\ \text Y & 2 & 2 & 3 & 3 & 10 \\ \text Z & 3 & 3 & 2 & 2 & 10 \\ \text Z & 4 & 4 & 1 & 1 & 10 \\ \hline \end{array}$ |
Kendall's coefficient of concordance is $0$.
Also known as
Some sources report Kendall's coefficient of concordance just as the coefficient of concordance.
Also see
- Results about Kendall's coefficient of concordance can be found here.
Source of Name
This entry was named for Maurice George Kendall.
Historical Note
Kendall's coefficient of concordance was introduced by Maurice George Kendall in $1939$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): coefficient of concordance
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): coefficient of concordance
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): concordance