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

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.

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.

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

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

One way ANOVA Sr. No. Group - A Group – B Group – C 1 8 10 15 2 11 3 7 9 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)]

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

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

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) = (8 + 10 + ...............12 + 10)2 = 2732 / 30 CF = G2 = 2732 = 2484.3 N 30

One Way ANOVA Group - A Group – B Group – C 8 10 15 11 7 9 6 5 14 4 13 12 16

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

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

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

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 – 2484.3 = 244.7

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

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

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

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 490 + 668.9 + 1440 = 2618.9

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 490 + 668.9 + 1440 = 2618.9 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 = 2618.9 – 2484.3 = 134.6

Step 4 - One Way ANOVA – Calculation of SSW Calculation of Sum of Square within the group (SSW) SSW = SST – SSA SSW = 244.7 – 134.6 = 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

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)]

Calculate ANOVA Sr. No. Arts Group Science Group M.P.Ed 1 15 12 2 14 3 11 10 4 13 5

One way ANOVA 6.82 Group 1 Group 2 Group 3 Group 4 15 20 10 30 13 24 22 12 9 29 26 8 21 27 7 25 28 18 3 14

Two – way ANOVA