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24

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 vari-
ance” 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.

Case Study: Comparison of Five Laboratories

The data shown in Table 24.1 (and in Table 20.1) were obtained by dividing a large quantity of prepared
material into 50 identical aliquots and having five different laboratories each analyze 10 randomly
selected specimens. By design of the experiment there is no real difference in specimen concentrations,
but the laboratories have produced different mean values and different variances. In Chapter 20 these
data were analyzed using a multiple t-test to compare the mean levels. Here we will use a one-way
ANOVA, which focuses on comparing the variation within laboratories with the variation between
laboratories. The analysis is one-way because there is one factor (laboratories) to be assessed.

One-Way Analysis of Variance
Consider an experiment that has k treatments (techniques, methods, etc.) with n t replicate observations
made under each treatment, giving a total of N = ∑kn t observations, where t = 1, 2,…,k. If the variation
within each treatment is due only to random measurement error, this within-treatment variance is a good
estimate of the pure random experimental error. If the k treatments are different, the variation between
the k treatments will be greater than might be expected in light of the variation that occurs within the
treatment.

© 2002 By CRC Press LLC