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.

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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.

• 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
• 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.

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