Statistics for Environmental Engineers

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Analysis of Variance to Compare k Averages

KEY WORDS ANOVA, ANOVA table, analysis of variance, average, between treatment variance, grand average, F test, F distribution, one-way ANOVA, sum of squares, within-treatment variance.

Analysis of variance (ANOVA) is a method for testing two or more treatments to determine whether their sample means could have been obtained from populations with the same true mean. This is done by estimating the amount of variation within treatments and comparing it to the variance between treatments. If the treatments are alike (from populations with the same mean), the variation within each treatment will be about the same as the variation between treatments. If the treatments come from populations with different means, the variance between treatments will be inflated. The “within variance” and the “between variance” are compared using the F statistic, which is a measure of the variability in estimated variances in the same way that the t statistic is a measure of the variability in estimated means.

Analysis of variance is a rich and widely used field of statistics. “… the analysis of variance is more than a technique for statistical analysis. Once understood, analysis of variance provides an insight into the nature of variation of natural events, into Nature in short, which is possibly of even greater value than the knowledge of the method as such. If one can speak of beauty in a statistical method, analysis of variance possesses it more than any other” (Sokal and Rohlf, 1969).

Naturally, full treatment of such a powerful subject has been the subject of entire books and only a brief introduction will be attempted here. We seek to illustrate the key ideas of the method and to show it as an alternative to the multiple paired comparisons that were discussed in Chapter 20.

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