This chapter gets off to a slow start. In the first section you are given some motivating examples of when comparing more than two population means might be important. In addition you see how multiple box plots can be a powerful graphical tool for comparisons among multiple populations.

The important information is in section 8.2. Here the basic methodology for extending the two population case to the multiple population case is developed. You will see how a test of differences among population means is transformed into a test of equality of two variance estimates. You are given measures of between mean and within mean variability. The null hypothesis is that there are no differences among the population means. If the null hypothesis is correct, the between and within mean variance estimates are actually measuring the same underlying parameter (the residual variance). The F-test for two variances, discussed in the last Unit, becomes the foundation for the test of equality of means. This is a very powerful tool that will be used over and over again in the remainder of the course. Take the time now to really understand what is done here and you will have little problem with the rest of the course.

The computations for performing the F-test for variances are organized into a table, referred to as the Analysis of Variance, or AOV (pronounced "A"-"Nova") table, or ANOVA (pronounced "An"-"Nova") table (Table 8.6 page 389). It is important that you learn how to compute this table and understand its parts. We will be developing ANOVA tables for increasingly complex situations in the rest of the course.

Next we take a short excursion into the realm of experimental design with a discussion of the simplest of all formal experimental designs, the Completely Randomized Design. Note that in this design we make some fairly strong assumptions about the underlying material to which the treatments are applied, assumptions that can be checked (see section 8.4). We also introduce the linear model (at the bottom of page 394) as a mathematical representation of what we expect to observe. Each treatment group becomes its own separate population. The hypothesis of equal treatment (population) means is formulated in the linear model as a hypothesis of equal treatment "effects" (the alpha values at the bottom of page 395).

When the assumptions for the linear model formulation of a Completely Randomized Design are not met, we typically explore a number of optional analyses. We first consider transformations of the data, typically the response variable, as is discussed in section 8.5. Alternatively we can change to a nonparametric test, an example of which is the Kruskal-Wallis test, given in section 8.6. Finally, there are other, more complex linear and nonlinear modeling approaches that will not be covered in this course.

The ANOVA for Testing Multiple Population Means (Powerpoint) and (PDF)

The Completely Randomized Design(Powerpoint) and (PDF)

AOV Assumptions and Data Transformations(Powerpoint) and (PDF)

The Kruskal-Wallis Test (Powerpoint) and (PDF)