15. Analysis of Variance

There are situations and experiments that require processes to be compared at more than two levels. Data from such experiments can be analysed using analysis of variance or ANOVA.

15.1. Comparing Population Means

There are other ways to compare population means than ANOVA, but these are based on the assumption of either paired observations or independent random samples, and can only be used to compare two population means. ANOVA can be used to compare more than two populations, and also uses assessments of variation, which forms a large problem in other methods.

15.2. One-Way ANOVA

The procedure for testing the equality of population means is called a one-way ANOVA. This procedure is based on the assumption that all included populations have a common variance.

The total sum of squares (SST) in this procedure is made up of a within-group sum of squares (SSW) and a between groups sum of squares (SSG): SST = SSW + SSG

This division of the SST forms the basis of the one-way ANOVA, as it expresses the total variability around the mean for the sample observations.

If the null hypothesis is true (all population means are the same) then both SSW and SSG can be used to estimate the common population variance. This is done by dividing by the appropriate number of degrees of freedom.

Because SSW and SSG both provide an unbiased estimate of the common population variance if the null hypothesis is true, a difference between the two values indicates that the null hypothesis is false. The test of the null hypothesis is thus based on the ratio of mean squares:

Where  and . With the assumptions that the population variances are equal and the population distributions are normal.

The closer the ratio is to 1, the less indication there is that the null hypothesis is false.

These results are also summarized in a one-way ANOVA table, which has the following format:

 Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-ratio Between groups SSG K – 1 MSG MSG/MSW

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