Psychology, 5th Edition by Robert A. Baron (eBook)
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?

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

Sample – already given as per table above, sampling not required.
Step 2 Sampling

Calculation of Means 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 
betweengroups 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 
withingroups 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 S^{2} F ratio Between groups 2 3.71 1.855 0.44 Within groups 18 75.43 4.19 Total N  1 79.14
Step 3 Calculate the necessary sample statistics.

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 4 Identify the region of rejection.

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.
Step 5 Make the statistical decision and form conclusions.
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