Question

Ten competitors in a music competition are ranked by three adjudicators in the following order.

Competitor

  Competitor

A

B

C

D

E

F

G

H

I

J

Adjudicator 1

 

Adjudicator 2

 

Adjudicator 3

1

 

3

 

6

6

 

5

 

4

5

 

8

 

9

10

 

4

 

7

3

 

7

 

1

2

 

10

 

2

4

 

2

 

3

9

 

1

 

10

7

 

6

 

5

8

 

9

 

8

 Use Spearman’s rank order correlation technique  to compute:

i)       The pair of adjudicators who have the nearest approach to common tastes in music festival and therefore give the higher correlation between them.                         (3 marks)

ii)     The pair of adjudicators who have the worst taste of music and post the worst correlation.                                                                                                                                 (3 marks)

iii)   The average correlation amongst the three adjudicators.                              (2 marks)

Answer 1

Let us calculate the spearman's coefficients-

$r=1-(\dfrac{6\sum d^2}{n^3-n})$

$\sum d^2_{12}=200\\\sum d^2{23}=204\\\sum d^2_{13}=64$

Then the coefficient of adjudicator 1 w.r.t 2

Cofficient $r_{12}=1-(\dfrac{6\times 200}{(10)^3-10})$

$=1-\dfrac{1200}{990}$

$=1-1.21=-0.21$

Then the coefficient of adjudicator 2 w.r.t 3

Cofficient $r_{23}=1-(\dfrac{6\times 204}{(10)^3-10})$

$=1-\dfrac{1224}{990}$

$=1-1.236=-0.236$

Then the coefficient of adjudicator 1 w.r.t 3

Cofficient $r_{13}=1-(\dfrac{6\times 64}{(10)^3-10})$

$=1-\dfrac{384}{990}$

$=1-0.837=0.163$

(i) As coefficient $r_{13}$ has positive value. So adjudicators 1 and 3 have the nearest approach to common taste in music festival and have highest correlation between them. 

(ii) Adjudicators 2 has worst taste of music.

and coefficient value= -0.236

(iii) Average correlation coefficient $=\dfrac{r_{12}+r_{23}+r_{13}}{3}$

$=\dfrac{-0.21-0.236+0.163}{3}=\dfrac{-0.283}{3}=-0.0943$