Kendall's Coefficient of Concordance/Examples/Arbitrary Example 2
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Example of Kendall's Coefficient of Concordance
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 $0$.
Proof
The mean $M$ is given by:
- $M = \dfrac 1 2 \times 4 \times \paren {4 + 1} = 10$
The $S$ value is given by:
\(\ds S\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \paren {s_i - M}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {10 - 10}^2 + \paren {10 - 10}^2 + \paren {10 - 10}^2 + \paren {10 - 10}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Hence Kendall's coefficient of concordance is shown to be:
\(\ds W\) | \(=\) | \(\ds \dfrac {12 S} {m^2 n \paren {n^2 - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {12 \times 0} {4^2 \times 4 \times \paren {4^2 - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$
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