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.

  • 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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IGNOU Solved Assignments: Q2 - MAPC MPC006 Statistics in Psychology - MPC006/ASST/TMA/2014-15
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