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increase our precision over using only the grand mean. Specifically, if the
independent variable explains some of the variance, that is, if the means
of the groups are significantly different, and if you know the group to
which a given participant belongs, then using the group mean rather than
the grand mean should improve our prediction precision. For example,
knowing that a given score was associated with a participant who took the
test in the blue room, we would predict on a better than chance basis that
the score was relatively higher than those of the participants who took the
test in the red room. This is treated as statistical explanation.
The prediction based on knowing the participants' group membership,
however,willnotbeperfect.Thisisbecause,aswecanseefromviewing
Table 3.1, the scores of the participants in each group differ from each other
to a certain extent. Knowing whether a student was associated with the
Blue Room or Red Room group can significantly increase the preciseness
of our prediction, but in the long run there will still be some error in
predicting the value of each student's score in this study. How much of
the total variability can be explained, as we will see later, can be indexed
by the proportion of the between-groups sum of squares with respect to
the total sum of squares.
3.4.3 WITHIN-GROUPS SUM OF SQUARES
As can be seen from the summary table in Table 3.2, the remaining source
of variance is associated with within-groups variance. The term
within
groups
provides a useful cue of what this portion of the total variance
represents: variability within each of the two groups. Consider the Red
Room group, whose mean on the dependent variable is 14. If room color
completely (100 percent) determined the score on the mood survey, then
each of the participants should have scored 14. Of course, they did not.
Therefore, factors other than room color affected their performance; per-
haps the participants were of different temperaments or different ages,
had different emotional histories, different genders, and so on. But which
combination of the wide range of possible factors was operating here
is unknown. Because of this, none of these variables can be statistically
analyzed to determine whether they are associated with the dependent
measure; to the extent that they are important, these other variables con-
tribute measurement variability to the study, and that unaccounted for
variability within each group is defined as
error variance
.
Note that these other “unknown” factors may be rather important
contributors to the differences in the scores that we observe. The point
is that we can statistically analyze the effects only of known (measured)
variables. Thus, if some important other variables were not assessed in
the study, they contribute to measurement error. A one-way between-
subjects design is limited to a single independent variable, and thus the
design limits the number of “known” effects to just one. Some of the more
complex designs that we will cover in this topic allow us to incorporate
