# Analysis of Variance - ANOVA - Numerical

The steps below explain the process of calculation of ANOVA. The general logic of analysis of variance is the same as that of significance tests.

## A study was conducted in which the teachers were asked to rate students for a particular trait on a ten point scale. With the help of data given below find out whether significant difference exists in the rating of the students by the teachers.

 Teachers Students A B C D E F G X 7 3 7 1 5 5 5 Y 5 5 9 4 3 5 4 Z 6 3 7 1 3 5 3

### How to calculate ANOVA?

#### Step 1 Formulate the statistical hypotheses and select a level of significance.

• For the given problem, Null Hypothesis, Ho is that no significant difference exists in the rating of the students by the teachers. Numerically, Ho → µx = µy = µz.
Number of groups = k = 3
Number of cases in each group = n = 7
Number of units in total = N = 21
Alternative Hypothesis, H1 is that there is significant difference in the rating of the students by the teachers.
Since, level of significance is not given we will assume it is α = 0.05, and also test at α = 0.01.

#### Step 2 Sampling

• Sample – already given as per table above, sampling not required.

#### Step 3 Calculate the necessary sample statistics.

•  S. No. Group X Group Y Group Z X X2 Y Y2 Z Z2 1 7 49 5 25 6 36 2 3 9 5 25 3 9 3 7 49 9 81 7 49 4 1 1 4 16 1 1 5 5 25 3 9 3 9 6 5 25 5 25 5 25 7 5 25 4 16 3 9 Total 33 183 35 197 28 138

Meanx = 33/7 = 4.71
Meany = 35/7 = 5
Meanz = 28/7 = 4
• Correction term: 𝐶𝑥=((∑〖𝑥𝑖 )^2 〗)/𝑁 = ((33+35+28)^2)/21 = 9216/21 = 438.86
• total sum of squares and degrees of freedom: 𝑆𝑆_𝑡𝑜𝑡𝑎𝑙 = ∑〖(∑〖𝑥_𝑖 )^2 〗〗−𝐶_𝑥 = (∑𝑋^2 +∑〖𝑌^2+∑〖𝑍^2)〗〗−𝐶_𝑥 = 183 + 197 + 138 – 438.86 = 79.14
Degrees of freedom = N - 1 = 21 – 1 = 20
• between-groups sum of squares, degrees of freedom, and variance estimate: 𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛=(∑〖(∑〖𝑥_𝑖 )^2 〗〗)/𝑁−𝐶_𝑥=((∑〖𝑋)^2 〗+(∑〖𝑌)^2+(∑〖𝑍)^2 〗〗)/𝑁−𝐶_𝑥 = 442.57 – 438.36 = 3.71
Degrees of freedom = k - 1 = 3 - 1 = 2
Mean square (variance estimate) = 𝑠_𝑏𝑒𝑡𝑤𝑒𝑒𝑛^2 = (𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛)/(𝑑𝑓_𝑏𝑒𝑡𝑤𝑒𝑒𝑛 ) = 3.71/2 = 1.855
• within-groups sum of squares, degrees of freedom, and variance estimate: 𝑆𝑆_𝑤𝑖𝑡ℎ𝑖𝑛 = 𝑆𝑆_𝑡𝑜𝑡𝑎𝑙−𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛 = 79.14−3.71 = 75.43
Degrees of freedom = N – k = 21 – 3 = 18
Mean square (variance estimate) = 𝑠_𝑤𝑖𝑡ℎ𝑖𝑛^2=(𝑆𝑆_𝑤𝑖𝑡ℎ𝑖𝑛)/(𝑑𝑓_𝑤𝑖𝑡ℎ𝑖𝑛 ) = 75.43/18 = 4.19
• F ratio: 𝐹 = (𝑠_𝑏𝑒𝑡𝑤𝑒𝑒𝑛^2)/(𝑠_𝑤𝑖𝑡ℎ𝑖𝑛^2 ) = 1.855/4.19 = 0.44
• Summary of Anova
 Source of variance df SS S2 F ratio Between groups 2 3.71 1.855 0.44 Within groups 18 75.43 4.19 Total N - 1 79.14

#### Step 4 Identify the region of rejection.

• With df (between) = 2, and df (within) = 18, the critical value of F as per the F tables is 3.55 at α = 0.05 and 6.01 at α = 0.01. These are the values of F beyond which the most extreme 5% and 1% of sample outcomes respectively fall if Ho is true.

#### Step 5 Make the statistical decision and form conclusions.

• Because the F ratio we calculated (0.44) is lower than the critical values of F at both α = 0.05 and α = 0.01, therefore, the obtained F ratio is found to be not significant at both α = 0.05 and α = 0.01.
• Therefore, we accept the null hypothesis, Ho, i.e., µx = µy = µz.
• No significant difference exists in the rating of the students by the teachers.

### If you found this useful, do remember to like Psychology Learners on Facebook and subscribe by email.

IGNOU Students - please visit the FREE IGNOU Help Center for latest updates.
IGNOU Solved Assignments: Q2 - MAPC MPC006 Statistics in Psychology - MPC006/ASST/TMA/2014-15