Question

Let $x_1,x_2,x_3,\dots ..,x_{}$ be the ranks of n individuals according to character $A$ and $y_1,y_2\dots ..y_n$ the ranks of the same individuals according to other character $B$ such that $x_i+y_i=n+1$ for $i=1,2,3\dots ..N$. Then the coefficient of rank correlation between the characters $A$ and $B$ is

• A. $1$
• B. $0$
• C. $-1$
• D. None of these

Answer 1

The correct option is C

$-1$

Explanation for the correct answer:

Given that,

$x_i+y_i=n+1fori=1,2,3\dots ..N$

Let $x_i–y_i=d_i$ then $2x_i=n+1+d_i$

$d_i=2x_i–(n+1)$

$$\begin{gathered} \sum _{i=1^n}d{i}^2=\sum _{i=1^n}[2x_i–{(n+1)}^2] \\ =\sum _{i=1^n}[x{i}^2+{(n+1)}^2–4(n+1)x_i] \\ =4\sum _{i=1^n}x{i}^2+{(n+1)}^2\left(n\right)–4(n+1)\sum x_i \\ =4\frac{[n(n+1)(2n+1\left)\right]}{6}+{(n+1)}^{2}n–4(n+1)\frac{[n(n+1\left)\right]}{2} \\ =\frac{n(n^2–1)}{3} \end{gathered}$$

Finding the value of $r.$

$$\begin{gathered} r=1–\frac{6\sum {d_i}^2}{n}(n^2–1) \\ =1-2 \\ =-1 \end{gathered}$$

Therefore, option (C) is the correct answer.