Discussion
\((X_{i}, Y_{i})\) \(i = 1, ......, 9\) is a random sample from bivariate population where all \(X_{i}\)'s are distinct and all \(Y_{i}\)'s are distinct \(i = 1, ......, 9\). If the number of concordance pairs is 9, the Kendall's sample correlation coefficient is :
Answer: B
Kendall's coefficient is \(-1/2\).
With 9 observations and no ties there are \(\binom{9}{2} = 36\) pairs, each concordant or discordant. With 9 concordant pairs, 27 are discordant, so
\[ \begin{aligned} \tau &= \frac{C - D}{\binom{n}{2}} \\ &= \frac{9 - 27}{36} = -\frac{1}{2} \end{aligned} \]
The sign is negative because discordant pairs outnumber concordant ones. +1/2 reverses the sign, and \(\pm 1/4\) come from dividing by 72 or using 9 alone instead of the difference.
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