Information Technology Reference
In-Depth Information
standard deviations of the groups. The logic for this is as follows. The
standard deviation represents the spread of scores around a mean. Think
of this score spread as a way of quantifying the “fuzzy boundary” of a
group mean. Now mentally place another group mean with its own fuzzy
boundary near the first group. If you place the second mean in such a
position that the two means with their fuzzy boundaries are visually quite
close, then in effect you cannot clearly distinguish between the two means.
If the second mean is placed further away from the first, then despite each
having a fuzzy boundary, it is possible to more clearly tell them apart.
Cohen's
d
is a way to quantify how clearly the means can be differ-
entiated. It takes into account the difference in value between the two
means with respect to the combined standard deviation of the groups. In
equation form, we have
Y
1
−
Y
2
=
SD
.
d
(4.4)
Not only did Cohen suggest this procedure to quantify effect size, he
provided general guidelines for interpreting values of
d.
He proposed that,
all else equal,
d
values of .2, .5, and .8 can be thought of as small, medium,
and large effect sizes, respectively. For example, if the mean difference
spans a distance of almost 1
SD
, then the means of the two groups can
be quite easily distinguished, and so we would judge the effect size to be
large.
Cohen extended his formulas to the situation in ANOVA where we have
more than two means. He used the symbol
f
to index the effect size for
multiple means and showed how
d, f,
and
2
are all related. Nonetheless,
many writers when referring to Cohen's effect size index will tend to focus
on
d
.
η
4.6 REPORTING THE RESULTS
Reporting the results of a statistically significant effect of the independent
variable involves presenting certain information to the reader. Here is
what you are obliged to provide:
The levels of the independent variable in plain English.
The means and standard deviations of the groups.
An indication, again in plain English, of the direction of the differ-
ence.
The
F
ratio and the degrees of freedom at which it was evaluated
(in this order: between-groups degrees of freedom, within-groups
degrees of freedom):
F
(1, 12)
=
18.75.
Whether or not the probability reached your alpha level:
p
<.
05.
The strength of the effect and the statistic used to make the evalua-
tion:
2
η
=
.
610.
