Question

In a dance competition the marks given by two judges to 10 participants are given below.

Participants	A	B	C	D	E	F	G	H	I	J
$1^{st}$ Judge	1	5	4	8	9	6	10	7	3	2
$2^{nd}$ Judge	4	8	7	6	5	9	10	3	2	1

Find the rank correlation coefficient.

Answer 1

When we are asked to find the rank correlation coefficient, we always find the Spearman’s ranks Correlation coefficient which gives the monotonic relationship between two variables unlike pearson correlation coefficient which gives linear relationship.
Formula to calculate the Spearman’s ranks Correlation coefficient = $R=1-\dfrac{6\sum{{{d}^{2}}}}{n\left( {{n}^{2}}-1 \right)}$
And, here d is the difference in two rank of each observation (i.e. $d=\left| {{R}_{1}}-{{R}_{2}} \right|$).

Complete step by step answer:
We can see from the question that we are required to calculate the rank correlation coefficient of the marks given by two judges in a dance competition.
So, to find the rank correlation coefficient we will use the concept of the Spearman’s ranks Correlation coefficient as it given the monotonic relationship between two variables.
And, we know that Spearman’s ranks Correlation coefficient is given by $R=1-\dfrac{6\sum{{{d}^{2}}}}{n\left( {{n}^{2}}-1 \right)}$ and here d is the difference between rank of two observations and is given as $d=\left| {{R}_{1}}-{{R}_{2}} \right|$
Since, we know that the marks given by two judges to 10 participants is given as:

Participants	A	B	C	D	E	F	G	H	I	J
$1^{st}$ Judge	1	5	4	8	9	6	10	7	3	2
$2^{nd}$ Judge	4	8	7	6	5	9	10	3	2	1

So, we say that for participant A difference between marks given by two judges is $d=\left| 1-4 \right|$ = 3
Similarly, for participant B difference between marks given by two judges is $d=\left| 5-8 \right|$ = 3
And, similarly for other participants we can calculate the difference between the marks given by two judges.
So, the table to calculate the Spearman’s ranks Correlation coefficient is:

Rank1(${{R}_{1}}$)(based on the marks of judge 1)| Rank2(${{R}_{2}}$)(based on marks of judge 2)| $d=\left| {{R}_{1}}-{{R}_{2}} \right|$| ${{d}^{2}}$
---|---|---|---
1| 4| 3| 9
5 | 8| 3| 9
4| 7| 3| 9
8 | 6| 2| 4
9| 5| 4| 16
6| 9| 3| 9
10| 10| 0| 0
7| 3| 4| 16
3| 2| 1| 1
2| 1| 1| 1
| | | $\sum{{{d}^{2}}=74}$

From the table we can say that $\sum{{{d}^{2}}=74}$.
Now, we will calculate the Spearman’s ranks Correlation coefficient using the formula $R=1-\dfrac{6\sum{{{d}^{2}}}}{n\left( {{n}^{2}}-1 \right)}$. Here, $\sum{{{d}^{2}}=74}$ and n is the total number of participants which is equal to 10.
So, $R=1-\dfrac{6\times 74}{10\left( {{10}^{2}}-1 \right)}$
Therefore, R = 1 – 0.44848
So, R = 0.5515
This is our required solution.
Hence, we can say that rank correlation coefficient is equal to 0.5515.

Note: Students are required to note that when we are asked to find the rank correlation coefficient then we have to calculate the Spearman’s ranks Correlation coefficient as it gives the monotonic relationship between two variables whereas pearson correlation coefficient which gives linear relationship. Also, students are required to memorize the formula to calculate the Spearman’s ranks Correlation coefficient and they should avoid calculation mistakes.