1. Analysis of Variance or ‘F’
Dr. Anshul Singh Thapa

2. An Introduction
The Analysis of Variance is an important method used to test the significance of the difference between 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.

3. The analysis of variance deals with the variance rather than with standard deviation or standard error.
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 deviation from the mean of the scores of a distribution.
In simple terms it is square of standard deviation.

4. Steps in calculation of ‘F’
Step 1 – Correction factor Step 2 – Sum of Square of Total SST Step 3 – Sum of Square between the Group SSA Step 4 – Sum of Square within the Group SSW Step 5 – Mean Sum of Squares between 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

5. Sum of Square within the groups Sum of Square among the group
Mean sum of Square among the groups (Variance among the groups)
F =
Mean sum of Square within the group (Variance within the groups)
Sum of Square within the groups
Mean sum of Square among the groups
(Variance among the group)
Sum of Square among the groups
Mean sum of Square within the group
(Variance within the Groups) = =
df (N – k)
df (k – 1)
Sum of Square within the groups
Total Sum of Square
Sum of Square among the group = + or =
Sum of Square within the groups
Total Sum of Square -
Sum of Square among the group =
Sum of Square among the group
Sum of Square of all the individual groups divided by n -
Correction Factor =
Total Sum of Square
Sum of Square of all the individual scores divided by N -
Correction Factor =
Correction Factor
Square of sum of all the individual scores divided by N

6. One way ANOVA Sr. No. Group - A Group – B Group – C 1 8 10 15 2 11 3 79 4 6 5 14 13 12 16
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)]

7. One Way ANOVA Group - A Group – B Group – C 8 10 15 11 7 9 6 5 14 4 1312 16
Sum: 70
Sum: 83
Sum: 120

8. Main Steps of One Way ANOVA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
Sum: 70
Sum: 83
Sum: 120
G (A+B+C) = 273

9. Step 1 - One Way ANOVA – Calculation of CF
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
Sum: 70
Sum: 83
Sum: 120
G (A+B+C) = 273
Calculation of Correction Factor (CF) = ( )2 = 2732 / 30 CF = G2 = 2732 = N 30

10. One Way ANOVA Group - A Group – B Group – C 8 10 15 11 7 9 6 5 14 4 1312 16

11. One Way ANOVA Group - A A2 Group – B B2 Group – C C2 8 64 10 100 15
225 11
121 7 49 9 81 6 36 5 25 14
196 4 16 13
169 12
144
256

12. Step 2 - One Way ANOVA – Calculation of SST
Group - A A2
Group – B B2
Group – C C2 8 64 10
100 15
225 11
121 7 49 9 81 6 36 5 25 14
196 4 16 13
169 12
144
256
Sum: 520
Sum: 709
Sum: 1500

13. Step 2 - One Way ANOVA – Calculation of SST
Group - A A2
Group – B B2
Group – C C2 8 64 10
100 15
225 11
121 7 49 9 81 6 36 5 25 14
196 4 16 13
169 12
144
256
Sum: 520
Sum: 709
Sum: 1500
Raw Sum of Square RSS (A2 + B2 + C2) = 2729

14. Step 2 - One Way ANOVA – Calculation of SST
Group - A A2
Group – B B2
Group – C C2 8 64 10
100 15
225 11
121 7 49 9 81 6 36 5 25 14
196 4 16 13
169 12
144
256
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 – = 244.7

15. Step 3 - One Way ANOVA – Calculation of SSA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16

16. Step 3 - One Way ANOVA – Calculation of SSA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
ΣA: 70
ΣB: 83
ΣC: 120

17. Step 3 - One Way ANOVA – Calculation of SSA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
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

18. Step 3 - One Way ANOVA – Calculation of SSA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
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 =

19. Step 3 - One Way ANOVA – Calculation of SSA
Group - A
Group – B
Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16
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 =
Calculation of Sum of Square among the groups (SSA) = SSA = (Summation of Sum of A divided by n plus Sum of B divided by n…….) – CF = SSA = – = 134.6

20. Step 4 - One Way ANOVA – Calculation of SSW
Calculation of Sum of Square within the group (SSW)
SSW = SST – SSA
SSW = – = 110.1
Step 5 - One Way ANOVA – Calculation of MSSA
Mean Sum of Square among the groups (MSSA)
MSSA = SSA/ k – 1
MSSA = 134.6/ 2 = 67.3
Step 6 - One Way ANOVA – Calculation of MSSW
Mean Sum of Square within the group MSSW
MSSW = SSW/ N – k
MSSW = 110.1/ 27 = 4.08
Step 7 - One Way ANOVA – Calculation of F ratio
F Ratio = MSSA/ MSSW = 67.3/ 4.08 = 16.50

21. Step 8 - One Way ANOVA Degree of Freedom: Among the Group = k – 1
Source of Variance df
Sum of Square
Mean Sum of Square
F - value
Among the groups 2
134.6
67.3
16.50*
Within the groups 27
110.1
4.08
*Significant at 0.05 level of Significance
Table value F.05 (2,27) = 3.35
Degree of Freedom:
Among 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)]

23. Calculate ANOVA Sr. No. Arts Group Science Group M.P.Ed 1 15 12 2 14 311 10 4 13 5

24. One way ANOVA 6.82 Group 1 Group 2 Group 3 Group 4 15 20 10 30 13 2422 12 9 29 26 8 21 27 7 25 28 18 3 14

25. Two – way ANOVA