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Analysis of Variance�or�‘F’

Dr. Anshul Singh Thapa

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An Introduction

  • The Analysis of Variance is an important method used to test the significance of the difference among the means.
  • The statistical procedure for testing variation among the means of more than two group is called analysis of variance (ANOVA).
  • The technique of analysis of variance was first devised by Sir Ronald Fisher, an English statistician who is also known as father of modern statistics.
  • The analysis of variance deals with the variance rather than with standard deviation or standard error.

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  • Before we go further the procedure and use of analysis of variance to test the significance difference between the means of various populations, it is essential first to have clear concept of the term variance.
  • In statistics the distance of scores from a central point i.e., Mean is called deviation and the index of variability is known as mean deviation or standard deviation. But sometimes the results may be somewhat more simply interpreted with the help of variance.
  • Variance of the sample is defined as the average of sum of the squared deviation from the mean of a distribution. In simple terms it is square of standard deviation.

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Steps in calculation of ‘F’

Step 1 – Correction factor

Step 2 – Sum of Square of Total (SST)

Step 3 – Sum of Square among the Groups (SSA)

Step 4 – Sum of Square within the Group (SSW)

Step 5 – Mean Sum of Squares among the groups (MSSA )

Step 6 – Mean Sum of Squares within the groups (MSSW)

Step 7 – F Ratio i.e., F = MSSA / MSSW

Step 8 – Summary of ANOVA

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One way ANOVA

Sr. No.

Group - A

Group – B

Group – C

1

8

10

15

2

10

8

11

3

7

7

9

4

6

8

8

5

5

9

14

6

4

6

13

7

9

7

12

8

6

9

16

9

7

11

12

10

8

8

10

k = 3 (Number of groups)

n = 10 (Number of subjects in each group)

N = 30 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

6 of 40

One Way ANOVA

Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

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One Way ANOVA

Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

Sum: 70

Sum: 83

Sum: 120

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Step 1 = Calculation of Correction Factor

Correction Factor = G2/N

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

10 of 40

Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

Sum: 70

Sum: 83

Sum: 120

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

Sum: 70

Sum: 83

Sum: 120

G (A+B+C) = 273

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

Sum: 70

Sum: 83

Sum: 120

G (A+B+C) = 273

Calculation of Correction Factor (CF) = G2 /N

(8 + 10 + ...............12 + 10)2 = 2732 / 30 = 2484.3

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Step 2 = Calculation of Sum of Square of Total

Sum of Square of Total = RSS – CF

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Group - A

​

Group – B

​

Group – C

​

8

​

10

​

15

​

10

​

8

​

11

​

7

​

7

​

9

​

6

​

8

​

8

​

5

​

9

​

14

​

4

​

6

​

13

​

9

​

7

​

12

​

6

​

9

​

16

​

7

​

11

​

12

​

8

​

8

​

10

​

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Group - A

A2

Group – B

​

Group – C

​

8

64

10

​

15

​

10

100

8

​

11

​

7

49

7

​

9

​

6

36

8

​

8

​

5

25

9

​

14

​

4

16

6

​

13

​

9

81

7

​

12

​

6

36

9

​

16

​

7

49

11

​

12

​

8

64

8

​

10

​

16 of 40

Group - A

A2

Group – B

B2

Group – C

​

8

64

10

100

15

​

10

100

8

64

11

​

7

49

7

49

9

​

6

36

8

64

8

​

5

25

9

81

14

​

4

16

6

36

13

​

9

81

7

49

12

​

6

36

9

81

16

​

7

49

11

121

12

​

8

64

8

64

10

​

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Group - A

A2

Group – B

B2

Group – C

C2

8

64

10

100

15

225

10

100

8

64

11

121

7

49

7

49

9

81

6

36

8

64

8

64

5

25

9

81

14

196

4

16

6

36

13

169

9

81

7

49

12

144

6

36

9

81

16

256

7

49

11

121

12

144

8

64

8

64

10

100

18 of 40

Group - A

A2

Group – B

B2

Group – C

C2

8

64

10

100

15

225

10

100

8

64

11

121

7

49

7

49

9

81

6

36

8

64

8

64

5

25

9

81

14

196

4

16

6

36

13

169

9

81

7

49

12

144

6

36

9

81

16

256

7

49

11

121

12

144

8

64

8

64

10

100

​

Sum: 520

​

Sum: 709

​

Sum: 1500

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Group - A

A2

Group – B

B2

Group – C

C2

8

64

10

100

15

225

10

100

8

64

11

121

7

49

7

49

9

81

6

36

8

64

8

64

5

25

9

81

14

196

4

16

6

36

13

169

9

81

7

49

12

144

6

36

9

81

16

256

7

49

11

121

12

144

8

64

8

64

10

100

​

Sum: 520

​

Sum: 709

​

Sum: 1500

Raw Sum of Square RSS (A2 + B2 + C2) = 2729

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Group - A

A2

Group – B

B2

Group – C

C2

8

64

10

100

15

225

10

100

8

64

11

121

7

49

7

49

9

81

6

36

8

64

8

64

5

25

9

81

14

196

4

16

6

36

13

169

9

81

7

49

12

144

6

36

9

81

16

256

7

49

11

121

12

144

8

64

8

64

10

100

​

Sum: 520

​

Sum: 709

​

Sum: 1500

Raw Sum of Square RSS (A2 + B2 + C2) = 2729

Calculation of Total Sum of Square (SST) = RSS – CF

SST = 2729 – 2484.3 = 244.7

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Step 3 =

Calculation of Sum of Square among the Groups

Sum of Square among the Groups =

ΣA2 / n + ΣB2 / n + ΣC2 / n – CF

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

ΣA: 70

ΣB: 83

ΣC: 120

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

ΣA: 70

ΣB: 83

ΣC: 120

ΣA2 / n = (70)2/10 = 490

ΣB2 / n = (83)2/10 = 688.9

ΣC2 / n = (120)2/10 = 1440

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

ΣA: 70

ΣB: 83

ΣC: 120

ΣA2 / n = (70)2/10 = 490

ΣB2 / n = (83)2/10 = 688.9

ΣC2 / n = (120)2/10 = 1440

490 + 668.9 + 1440 = 2618.9

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Group - A

Group – B

Group – C

8

10

15

10

8

11

7

7

9

6

8

8

5

9

14

4

6

13

9

7

12

6

9

16

7

11

12

8

8

10

ΣA: 70

ΣB: 83

ΣC: 120

ΣA2 / n = (70)2/10 = 490

ΣB2 / n = (83)2/10 = 688.9

ΣC2 / n = (120)2/10 = 1440

490 + 668.9 + 1440 = 2618.9

SSA = (Summation of Sum of A2 divided by n plus Sum of B2 divided by n…) – CF

= SSA = 2618.9 – 2484.3 = 134.6

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Step 4 - Calculation of SSW

Calculation of Sum of Square within the group (SSW)

SSW = SST – SSA

SSW = 244.7 – 134.6 = 110.1

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Step 5 - Calculation of MSSA

Mean Sum of Square among the groups (MSSA)

MSSA= SSA/ k – 1

MSSA = 134.6/ 2 = 67.3

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Step 6 - Calculation of MSSW

Mean Sum of Square within the group MSSW

MSSW = SSW/ N – k

MSSW = 110.1/ 27 = 4.08

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Step 7 - Calculation of F ratio

F Ratio = MSSA/ MSSW = 67.3/ 4.08 = 16.50

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

​

​

​

​

​

​

​

​

​

​

​

​

​

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

​

​

​

​

Within the groups

​

​

​

Total

​

​

​

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

2

​

​

​

Within the groups

27

​

​

Total

29

​

​

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

2

134.6

​

​

Within the groups

27

110.1

​

Total

29

244.7

​

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

2

134.6

67.3

​

Within the groups

27

110.1

4.08

Total

29

244.7

​

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Step 8 – ANOVA Summary

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

2

134.6

67.3

16.50*

Within the groups

27

110.1

4.08

Total

29

244.7

​

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Step 8 – ANOVA Summary

*Significant at 0.05 level of Significance

​

Source of Variance

df

Sum of Square

Mean Sum of Square

F - value

Between the groups

2

134.6

67.3

16.50*

Within the groups

27

110.1

4.08

Total

29

244.7

​

Table value F.05 (2,27) = 3.35

​

Degree of Freedom:

Between the Group = k – 1

Within the group = N – k

k = 3 (Number of groups)

n = 10 (Number of subjects in each group)

N = 30 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

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39 of 40

Calculate ANOVA

Sr. No.

Arts Group

Science Group

M.P.Ed

1

15

12

12

2

14

14

15

3

11

10

14

4

12

13

10

5

10

11

10

40 of 40

One way ANOVA

Group 1

Group 2

Group 3

Group 4

15

20

10

30

10

13

24

22

12

9

29

26

8

22

12

20

21

24

27

29

7

25

21

28

13

18

25

25

3

12

14

15