Introduction to ANOVA

We have previously discussed methods of testing whether

1. Two populations have equal means (independent two-sample tests).
2. The means from two measurements on one population are the same (paired tests).

Analysis of Variance (ANOVA) allows us to extend this to more than two populations or measurements (treatments/). That is, we can test the following:

1. Are all the means from more than two populations equal?
2. Are all the means from more than two treatments on one population equal? (This is equivalent to asking whether the treatments have any overall effect.)

To set our notation, let I be the number of populations or treatments being compared and let be the I means. Then the hypotheses for testing are

To test these hypotheses, we require a random sample from each population or treatment.

NOTE:\ For computational purposes, the ANOVA equations for the multiple population case and the multiple treatment on one population case are the same. However, the interpretation of hypotheses and results is slightly different. Thus,

1. Multiple populations: is the true mean of population i.
2. Multiple treatments: is the true average response when treatment i is applied.