Coefficient of determination, Mathematics

It refers to the ratio of the explained variation to the total variation and is utilized to measure the strength of the linear relationship. The stronger the linear relationship the closer the ratio will be to one.

Coefficient determination = Explained variation/Total variation

Illustration of Rank Correlation Coefficient

In a beauty competition two assessors were asked to rank the 10 contestants by using the professional assessment skills. The results obtained were described as shown in the table below as:

Contestants

1st assessor

2nd assessor

A

6

5

B

1

3

C

3

4

D

7

6

E

8

7

F

2

1

G

4

8

H

5

2

J

10

9

K

9

10

REQUIRED

Compute the rank correlation coefficient and thus comment briefly on the value acquired

 

 

 

d

d2

A

6

5

1

1

B

1

3

-2

4

C

3

4

-1

1

D

7

6

1

1

E

8

7

1

1

F

2

1

1

1

G

4

8

-4

16

H

5

2

3

9

J

10

9

+1

1

K

9

10

-1

1

 

 

 

 

Σ d2 = 36

 

The rank correlation coefficient R

            R = 1 -  {(6Σd2)/(n(n2 -1))}

                        = 1 -  {(6(36))/(10(102 -1))}

                        = 1 -  (216/990)

                        = 1 - 0.22

                        = 0.78

Comment: because the correlation is 0.78 it implies that there is high positive correlation among the ranks awarded to the contestants. 0.78 > 0 and 0.78 > 0.5

Illustration

Contestant

1st assessor

2nd assessor

d

d2

A

1

2

-1

1

B

5 (5.5)

3

2.5

6.25

C

3

4

-1

1

D

2

1

1

1

E

4

5

-1

1

F

5 (5.5)

6.5

-1

1

G

7

6.5

-0.5

0.25

H

8

8

0

0

 

 

 

 

Σ  d2 = 11.25

Required: Complete the rank correlation coefficient

∴ R = 1 - {(6Σd2)/(n(n2 -1))}

 

= 1 -  {(6(11.25))/(8(63))}

= 1 -  (67.5/507)

= 1 - 0.13

= 0.87

This shows high positive correlation

Posted Date: 2/18/2013 5:27:44 AM | Location : United States







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