It should be noted that this example is probably representative
of how ANOVA is used in practice, but it has a small theoretical
problem, in that ANOVA is designed to answer a
question different from the one being asked here. Namely, in this
example we are trying to decide whether three samples, all from the
same population group -- the
student population from which the subjects were drawn -- have
significantly different averages. (That is what we want to know, of
course, because if they were different to begin with, comparing the
results of later instruction by different methods would not be fair).
But ANOVA answers the question: given samples taken from several
different groups, having different sample averages (as they
almost certainly will), are those averages different enough to
conclude that the groups from which the samples were
taken have different averages? Since our "three groups" in this
example are all the same group, of course the group averages are
identical. Compare this to one of our examples of 2-sample
z-tests from the text (of which ANOVA is a generalization
to more groups; in the case of two groups,
F is just the square of our usual z, or more precisely
of t): We had small samples of Colgate and Hamilton students
take a standardized math test to decide whether the average math
ability of all Colgate students differed from the average
ability of all Hamilton students -- the groups were all students
at each college, the samples the ones who took the tests.
Return to the discussion of the
example.