Summary comments on the nature of ANALYSIS OF
VARIANCE:
 The technique tests for significant differences
between the means of k groups based on two
different estimates of population VARIANCE.
 One estimate is based on the variance
within each of k samples.
 The other is based on the variance between
(among) the sample means.
 The F test is simply a ratio between these two
estimates:
F =

estimate of variance based on BETWEEN mean
differences

estimate of variance based on WITHIN group
differences

 If this ratio is small, then the two estimates tend
to agree, and we conclude that the observed differences
in means reflect differences allowable by drawing random
samples from the same population.
 If the ratio is "large," however, we conclude
that the differences among our groups do not simply
reflect random error in sampling.

Interpretation of the F ratio:
 An F ratio less than 1 is never signficant for
rejecting the null hypothesis.
 That would show more variation within groups
than between groups, and thus the groups would
explain no variation at all in X.
 Ratios larger than 1 may be significant:
one must find out by checking the table of the F
distribution corresponding to the chosen alpha
level.
 Degrees of freedom (df) for the "between"
estimate (the larger) run along the top of the F
table
 df for the "within" estimate run down the
side.
 An observed value larger than the one in the table
means the difference is significant at that alpha
value.
 Relationship of F to t:
 Conceptually: F is a generalization of
Ttest for two groups
 Computationally:
 When there are two groups (and thus df BSS =
1)
 t = square root of F (or t^{2} =
F)

Analysis of variance with SPSS:
 SPSS Programs for analysis of variance
 Under the Analyze Menu, choose Compare
Means
 Then select OneWay ANOVA
 Press the "Options" button and check
"Descriptive" and "Means plot"
 Transfer your dependent variable
into the "Dependent List"
 Transfer your independent
(discrete) variable into the
"Factor" list
 Oneway analysis of variance programs follow a common
form, reading from right to left:

Analysis of variance in research: a simple
example:
 Lacy, "Political Knowledge of College Activist
Groups: SDS, YAF, and YD"
 Intellectual Problem: previous research has
shown that liberal students were better informed than
nonliberals
 But this research compared activist and
nonactivist students
 The conservative activist students need to be
studied too
 Existing studies suggest no difference in
intelligence between these two groups
 Lacy's Research Design and Data Analysis
 Studied 15 YAF, 39 SDS, and 33 YD at University of
Houston
 Compared them on test of political knowledge
 Analysis of variance for knowledge of American
government showed no significant differences among the
group means
