```                      ANALYSIS OF VARIANCE

I.  Introduction.
A.  In independent measures t-tests, the means of two
independent samples (groups) on some interval variable
were compared with the t-test indicating whether the
means of the groups were different. This is fine when you
only have two groups, such as with gender. However, when
you have more than two groups, a different type of test
is required.
B.  Analysis of variance (ANOVA) is a statistical test in
which the means of 3 or more groups on some interval
level variable can be compared simultaneously.  It
indicates whether at least one of the group means is
significantly different from the others.
C.  In analysis of variance, the total variance in the
combined sample (all groups taken together) is
partitioned (divided) into two parts.
1.  The first part represents the amount of variation
that exists within each of the the groups being
compared.
2.  The second part represents the amount of variation
between the groups being compared.
3.  The ratio of the variation between the groups to
the variation within the groups is called an
F-ratio.  The F-ratio is used in Anova to determine
if the means of the groups being compared can be
considered equal.

II.  Doing analysis of variance.
A.  State the null hypothesis.
B.  Choose a statistical test.
C.  Check assumptions.
1.  Three or more independent random probability
samples.
2.  Interval-ratio variable being compared across three
or more categories defined by a categorical
variable.
3.  Populations are normally distributed.
4.  Population variances are equal.
a.  Sample size of each group is relatively large
(30 or more).
b.  Sample sizes of the groups approximately
equal.
D.  Select an alpha level.
E.  Calculating the F-ratio and making a decision about the
null hypothesis.
1.  Calculate the statistic.
2.  Determine the probability associated with the
statistic.
3.  Compare probability to alpha level.

III.  Limitations and extensions of the one-way analysis of variance.
A.  The analysis of variance test just performed is called a
one-way analysis of variance because means are being
compared across a single set of groups or categories.
Extensions anova also allow for comparisons across
several different sets of groups or categories, called
multiple analysis of variance or MANOVA, and for the
effects of interval variables (besides the dependent
variable) to be controlled, called analysis of
covariance or ANCOVA.
B. Analysis variance only tells you if at least one of the
groups being compared differs from the others.  It does not
tell you which groups differ from each other.  However, there
are additional tests that can be performed that do indicate
such differences.  These are called multiple comparison
tests.
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