Question

From a random sample space, $5$ students have been selected. Their marks in Maths and Statistics are given below. The rank correlation coefficient is

Roll number	1	2	3	4	5
Marks in maths	85	60	73	40	90
marks in statistics	93	75	65	50	80

• A. $0.8$
• B. $0.6$
• C. $0.5$
• D. $-0.8$

Answer 1

The correct option is A

$0.8$

Explanation for the correct option:

Step 1. Find the rank correlation coefficient of the given data:

Roll number	Math	Statistic	$x$(Rank of math)	$y$(Rank of statistic)
1	85	93	2	1
2	60	75	4	3
3	73	65	3	4
4	40	50	5	5
5	90	80	1	2

Step 2. Evaluate the difference in rank and its square:

Roll number	Math	Statistic	$x$(Rank of math)	$y$(Rank of statistic)	$d$(Difference in rank:$x-y$)	$d^2$(Square of the difference)
1	85	93	2	1	1	1
2	60	75	4	3	1	1
3	73	65	3	4	1	1
4	40	50	5	5	0	0
5	90	80	1	2	1	1

Step 3. Find the sum of square of the difference in the rank $\left(x-y\right)$:

$\sum _{}{d}^2=1+1+1+0+1$

$=4$

As we know, the rank correlation coefficient $\mathit{\rho }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{-}\frac{\mathbf{6}{\displaystyle \mathbf{\sum }_{}}{\mathbf{d}}^{\mathbf{2}}}{\mathbf{n}\mathbf{(}{\mathbf{n}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}}$

where, $d$ is the difference in ranks

and $n$ is the total number of observations

Step 4. Substitute the values of $d$ and $n$ in the formula:

$\rho =1-\frac{6\times 4}{5(5^2-1)}$

$=1-\frac{24}{5\left(25-1\right)}$

$=1-\frac{24}{5\times 24}$

$=1-\frac{1}{5}$

$=\frac{4}{5}$

$=0.8$

Thus, the rank of correlation coefficient of the given observations is $\mathbf{0}\mathbf{.}\mathbf{8}$

Hence, Option ‘A’ is Correct.