Question

Two numbers within the brackets denote the ranks of $10$ students of a class in two subjects $(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)$, then rank correlation coefficient is

• A. $0$
• B. $-1$
• C. $1$
• D. $0.5$

Answer 1

The correct option is B

$-1$

Explanation for correct option
On relating the difference of ranks between the pairs $1\&10,2\&9,3\&8,4\&7,5\&6...$students.

The data shows in the given table

$x$	$y$	$d=x-y$	$d^2$
$1$	$10$	$-9$	$81$
$2$	$9$	$-7$	$49$
$3$	$8$	$-5$	$25$
$4$	$7$	$-3$	$9$
$5$	$6$	$-1$	$1$
$6$	$5$	$1$	$1$
$7$	$4$	$3$	$9$
$8$	$3$	$5$	$25$
$9$	$2$	$7$	$49$
$10$	$1$	$9$	$81$
			$\sum _{}{d}^2=330$

$\therefore$Rank correlation $=1-\frac{6\sum _{}{d}^2}{n(n^2-1)}$
$$\begin{gathered} =1-\frac{6\times 330}{10({10}^2-1)}\left[\because n=10\right] \\ =1-\frac{198}{99} \\ =1-2 \\ =-1 \end{gathered}$$

Hence option(B) is correct