Kendall's Coefficient of Concordance/Examples/Arbitrary Example 1

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Example of Kendall's Coefficient of Concordance

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 $\dfrac 1 {16}$.


Proof

The mean $M$ is given by:

$M = \dfrac 1 2 \times 4 \times \paren {3 + 1} = 8$

The $S$ value is given by:

\(\ds S\) \(=\) \(\ds \sum_{i \mathop = 1}^n \paren {s_i - M}^2\)
\(\ds \) \(=\) \(\ds \paren {7 - 8}^2 + \paren {9 - 8}^2 + \paren {8 - 8}^2\)
\(\ds \) \(=\) \(\ds 2\)


Hence Kendall's coefficient of concordance is shown to be:

\(\ds W\) \(=\) \(\ds \dfrac {12 S} {m^2 n \paren {n^2 - 1} }\)
\(\ds \) \(=\) \(\ds \dfrac {12 \times 2} {4^2 \times 3 \times \paren {3^2 - 1} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {16}\)

$\blacksquare$


Sources

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