Question

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Explain the seeming contradiction in the One-Way ANOVA; namely, that the null hypothesis is about comparing the means of three or more populations, whereas the actual testing of means is about using ANOVA analysis to compare variances. Why is this so?

Answer 1

The analysis of variance is a powerful statistical tool for tests of significance.The test of significance based on t-distribution is an adequate procedure only for testing the significance of difference between two sample means.In a situation when we have three or more samples to consider at a time an alternative procedure is needed .And that procedure is technique of analysis of variance.

The basic purpose of the analysis of variance is to test the homogeneity of several means.

This technique is not designed to test the equality of several population variances.

since in any analysis variation is inherent in nature.The variation in any set of numerical data is due to a number of causes which may be classified as...I) Assignable causes ii) chance causes.

The variation due to Assignable causes can be detected and measured but the variation due to chance causes is beyond the control of human hand and cannot be traced separately.

ANOVA is "separation of variance ascribable to one group of causes from the variance ascribable to other group".By this technique the total variation in sample data is expressed as the sum of its non negative components where each of these components is a measure of the variation due to some specific independent source or factor or cause.

ANOVA estimate the amount of variation due to each independent factors separately and then compare these estimates due to Assignable factors with the estimates due to chance factors.

Hence the main objective of ANOVA is to examine if there is significant difference between the class means in view of inherent variability within the separate classes.