Question

Calculate the rank correlation coefficient between marks assigned to 10 students by judges x and y in a certain competitive exam where marks of by judge x is 52,53,42,60,45,41,37,38,25,27 and marks by judge y is 65,68,43,38,77,48,35,30,25,0

Answer 1

Answer:

The rank correlation coefficient between marks assigned to 10 students by judges $x$ and $y$ is 0.746.

Step-by-step explanation:

Given information:

$\text{Judge} \, x \,\, |52 \,\, 53 \,\, 42 \,\, 60 \,\, 45 \,\, 41 \,\, 37 \,\, 38 \,\, 25 \,\, 27|\\\text{Judge} \, y \,\, |65 \,\, 68 \,\, 43 \,\, 38 \,\, 77 \,\, 48 \,\, 35 \,\, 30 \,\, 25 \,\, \,\,0\,|$

Concept Used:

The rank correlation coefficient is denoted by $r_R$ or $\rho$

$r_R=\rho=1-\dfrac{6\sum{D^2}}{N^3-N}$ where $N$ represents the number of observations and $D$ represents the difference between the $x$ rank and the $y$ rank for each pair of data.

Solution

Step 1 of 4:

First, calculate the rank of Judge $x$.

$\text{X} \,\,\,\, |\,\,\,52 \,\,\,\,\,\,\,\, 53 \,\,\,\,\,\,\,\, 42 \,\,\,\,\,\,\,\, 60 \,\,\,\,\,\,\,\, 45 \,\,\,\,\,\,\,\, 41 \,\,\,\,\,\,\, 37 \,\,\,\,\,\,\,\, 38 \,\,\,\,\,\,\,\, 25 \,\,\,\,\,\,\,\,\, 27|\\\text{R}_1 \, |\,\,\, 3 \,\,\,\,\,\,\,\,\,\,\,\,\, 2 \,\,\,\,\,\,\,\,\,\,\, 5 \,\,\,\,\,\,\,\,\,\, 1 \,\,\,\,\,\,\,\,\,\,\, 4 \,\,\,\,\,\,\,\,\,\,\, 6 \,\,\,\,\,\,\,\,\,\, 8 \,\,\,\,\,\,\,\,\,\,\,\, 7 \,\,\,\,\,\,\,\,\,\, 10 \,\,\,\,\,\,\,\,\, 9\,\,|$

Step 2 of 4:

Secondly, calculate the rank of Judge $y$.

$\text{Y} \, \, \,\, |\,\,\,65 \,\,\,\,\,\,\,\, 68 \,\,\,\,\,\,\,\,\, 43 \,\,\,\,\,\,\,\,\,\,\, 38 \,\,\,\,\,\,\,\,\, 77 \,\,\,\,\,\,\,\, 48 \,\,\,\,\,\,\,\,\, 35 \,\,\,\,\,\,\, 30 \,\,\,\,\,\,\, 25 \,\, \,\,0|\\\text{R}_2\,|\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,\,7\,\,\,\,\,\,\,\,\,\,8\,\,\,\,\,\,\,\,\,\,9\,\,\,10|$

Step 3 of 4:

Now, calculate the value of $D=R_1-R_2$.

$\text{D}=\text{R}_1-\text{R}_2\,\,\,\,\,\,\,|\,\,0\,\,\,\,\,0\,\,\,\,0\,\,\,-5\,\,\,\,\,\,3\,\,\,\,\,\,\,2\,\,\,\,\,\,\,1\,\,\,\,\,\,1\,\,\,\,\,\,1\,\,\,\,-1|\\D^2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,0\,\,\,\,\,0\,\,\,\,0\,\,\,\,\,\,\,25\,\,\,\,\,\,9\,\,\,\,\,\,\,4\,\,\,\,\,\,\,1\,\,\,\,\,\,1\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,1|$

Calculate $\sum{D^2}$.

$\sum{D}^2=42$

Step 4 of 4:

Now, calculate the rank correlation coefficient between marks assigned to 10 students by judges $x$ and $y$.

$r=1-\dfrac{6\sum{D^2}}{N^3-N}\\\\=1-\dfrac{6\cdot42}{10^3-10}\\\\=1-\dfrac{252}{990}\\\\=1-0.254\\\\=0.746$

Hence, the rank correlation coefficient between marks assigned to 10 students by judges $x$ and $y$ is 0.746.

Answer 2

Answer:

Hence, the rank correlation coefficient between marks assigned to 10 students by judges X and Y is 0.745.

Step-by-step explanation:

Given the marks scored by 10 students given by two judges X and Y

$\begin{array}{ccccccccccc}X&52&53&42&60&45&41&37&38&25&27\\Y&65&68&43&38&77&48&35&30&25&0\end{array}\right]$

Giving the ranks to the scores given by both the Judges  and arranging them in tabular column

$\left|\begin{array}{c}S.No.&1&2&3&4&5&6&7&8&9&10\end{array}\right \left|\begin{array}{c}X&52&53&42&60&45&41&37&38&25&27\9\end{array}\right| \left\begin{array}{c}Y&65&68&43&38&77&48&35&30&25&0\end{array}\right\left|\begin{array}{c}R_{X}&3&2&5&1&4&6\\8&7&10&9\end{array}\right|\left\begin{array}{c}R_{X}&3&2&5&6&1&4&7&8&9&10\end{array}\right|\begin{array}{c}d=R_{X}-R_{Y}&0&0&0&-5&3&2&1&-1&1&-1\end{array}\right|\left| \begin{array}{c}d^{2} &0&0&0&25&9&4&1&1&1&1\end{array}\right|$  

The rank correlation coefficient for $n$ number of observations with the difference between the ranks $d$ in each pair of data is given by  

$r=1-\frac{6\Sigma d^{2} }{n(n^{2}-1 )}$

sum of $d^{2} }$ values is $\Sigma d^{2} =42$

$\therefore r=1-\frac{6\times 42 }{10((10)^{2}-1 )}$

$=1-\frac{6\times 42 }{10\times 99}$

$=1-\frac{ 14 }{5\times 11}$

$=\frac{ 41 }{55}=0.745$

Hence, the rank correlation coefficient between marks assigned to 10 students by the two judges is 0.745.