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Coefficient of Determination Formula

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  • Last Updated : 15 May, 2022

Coefficient of determination is defined as the fraction of variance predicted by the independent variable in the dependent variable. It shows the degree of variation in the data collection offered. It is also known as R2 method which is used to examine how differences in one variable may be explained by variations in another. It is used in statistical analysis to predict and explain the future events of a model. It is proportional to the square of the correlation and its value lies between 0 and 1. If its value is zero, the dependent variable cannot be predicted based on the independent variable. If it is 1, the dependent variable may be predicted without mistake from the independent variable. And if it is between 0 and 1, it reflects how well the dependent variable can be predicted. Its value is equal to the square of the correlation coefficient, that is, r2.

Formula

r^2=\left[\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{\left [ n\sum x^{2}-(\sum x)^{2} \right ]\left [ n\sum y^{2}-(\sum y)^{2} \right ]}}\right]^2

where,

r2 is the coefficient of determination,

n is the number of observations of data set,

Σx is the sum of the first variable,

Σy is the sum of the second variable,

Σxy is the sum of the product of first and second variable,

Σx2 is the sum of the squares of the first variable,

Σy2 is the sum of the squares of the second variable.

If residual sum of squares and total sum of squares of data values are given, the formula for coefficient of determination is given by,

r2 = 1 – (R/T)

where,

r2 is the coefficient of determination,

R is the residual sum of squares,

T is the total sum of squares.

Sample Problems

Problem 1. Calculate the coefficient of determination for the data:

x

y

1

4

4

8

6

9

8

10

Solution:

The given data set is,

x

y

xy

x2

y2

1

4

4

1

16

4

8

12

16

64

6

9

15

36

81

8

10

18

64

100

Σx = 19

Σy = 31

Σxy = 49

Σx2 = 117

Σy2 = 261 

Using the formula we get,

r2 = [ (4 (49) – (19) (31)) / ((4 (117) – 361) (4 (261) – 961)) ]2

= (-393/8881)2

= (-0.044)2

= 0.001936

Problem 2. Calculate the coefficient of determination for the data:

x

y

5

3

2

8

4

1

7

5

The given data set is,

x

y

xy

x2

y2

5

3

15

25

9

2

8

16

4

64

4

1

4

16

1

7

5

35

49

25

Σx = 18

Σy = 17

Σxy = 70

Σx2 = 94

Σy2 = 99

Using the formula we get,

r2 = [ (4 (70) – (18) (17)) / ((4 (94) – 324) (4 (99) – 289)) ]2

= (-26/5564)2

= (-0.046)2

= 0.002116

Problem 3. Calculate the coefficient of determination if correlation coefficient is 0.5.

Solution:

We have,

r = 0.5

Using the formula we get,

Coefficient of determination = r2

= (0.5)2

= 0.25

Problem 4. Calculate the coefficient of determination if correlation coefficient is 0.82.

Solution:

We have,

r = 0.82

Using the formula we get,

Coefficient of determination = r2

= (0.82)2

= 0.67

Problem 5. Calculate the correlation coefficient if the coefficient of determination is 0.54.

Solution:

We have,

r2 = 0.54

Using the formula we get,

Coefficient of correlation = √r2

= √0.54

= 0.734

Problem 6. Calculate the correlation coefficient if the coefficient of determination is 0.68.

Solution:

We have,

r2 = 0.68

Using the formula we get,

Coefficient of correlation = √r2

= √0.68

= 0.82

Problem 7. Calculate the coefficient of determination if the residual sum of squares is 100 and total sum of squares is 200.

Solution:

We have, 

R = 100

T = 200

Using the formula we get,

r2 = 1 – (R/T)

= 1 – (100/200)

= 1 – 1/2

= 1/2

= 0.5


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