Question

Let $x_{1},x_{2},x_{3},.....,x_{n}$ be the rank of n individuals according to character A and $y_{1},y_{2},......,y_{n}$ the ranks of same individuals according to other character B such that $x_{i} + y_{i} = n + 1$ for $i = 1,2,3,.....,n$. Then the coefficient of rank correlation between the characters A and B is

• A. 1
• B. 0
• C. – 1
• D. None

Answer 1

Answer: (C)

– 1

Answer 2

$x_{i} + y_{i} = n + 1$ for all $i = 1,2,3,.....,n$

Let $x_{i} - y_{i} = d_{i}$. Then, $2x_{i} = n + 1 + d_{i}$ ⇒ $d_{i} = 2x_{i} - (n + 1)$

$\therefore$ $\sum_{i = 1}^{n}{d_{i}}^{2} = \sum_{i = 1}^{n}{\lbrack 2x_{i} - (n + 1)\rbrack^{2}}$ = $\sum_{i = 1}^{n}{\lbrack 4x_{i}^{2} + (n + 1)^{2} - 4x_{i}(n + 1)\rbrack}$

$\sum_{i = 1}^{n}{d_{i}}^{2} = 4\sum_{i = 1}^{n}{{x_{i}}^{2} + (n)(n + 1)^{2} - 4(n + 1)\sum_{i = 1}^{n}x_{i}}$

= $4 \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + ( n ) ( n + 1 ) ^ { 2 } - 4 ( n + 1 ) \frac { n ( n + 1 ) } { 2 }$ $\sum_{i = 1}^{n}{d_{i}}^{2} = \frac{n(n^{2} - 1)}{3}$

$\therefore$ $r = 1 - \frac{6\sum_{}^{}d_{i}^{2}}{n(n^{2} - 1)} = 1 - \frac{6(n)(n^{2} - 1)}{3(n)(n^{2} - 1)}$ i.e., $r = - 1$.