Download Cohen`s d, Cohen`s f, and η2

Cohen’s d, Cohen’s f, and 2
Cohen’s d , the parameter, is the difference between two population means divided by their
common standard deviation. Consider the Group 1 scores in dfr.sav. Their mean is 3. The sum of
the squared deviations about the mean is 9.0000. Since there are nine scores, the population
variance is 9/9 = 1, and the standard deviation is 1. Group 2 has the same number of scores, sum of
squared deviations about the mean, variance and standard deviation, but a mean of 4. Cohen’s d is
(4-3)/1 = 1. By Cohen’s benchmarks for d, this is a large effect.
Cohen’s f, the parameter, is the standard deviation of the population means divided by their
(3  3.5)2  (4  3.5)2
 0.25, yielding a
2
standard deviation of .5. Cohen’s f is .5/1 = .5. For the two population case, a d of 1 is equivalent to
an f of .5. By Cohen benchmarks for f, .5 is a large effect.
common standard deviation. The variance of 3 and 4 is
If you hate arithmetic, you can use G*Power to calculate the value of f. Select F, ANOVA,
fixed effects, omnibus, one way, a priori, enter the means and standard deviations, Calculate:
Cohen_d_f_r
2
If you conduct a simple linear regression relating the scores in dfr.sav to group membership,
you will obtain  = .447, a close to large effect by Cohen’s benchmarks for rho. The proportion of
variance in the scores explained by group membership is .4472 = .20. This is a squared point-biserial
correlation coefficient, but is more commonly referred to as eta-squared.
2
With equal population sizes, the relationship between f and  is f 
. For our
1  2
2
populations, that is
.20
= .5.
1  .20
Suppose we have three populations, as in drf3.sav. The means are 3, 4, and 5, and each
(3  4)2  (4  4)2  (5  4)2
 2/3, yielding a
3
standard deviation of .8165. Cohen’s f is .8165/1 = .8165.
within-population variance is 1. The variance of 3, 4, 5 is
The among groups sum of squares for these data is 18 and the total sum of squares is 45, for
an  of 18/45 = .40. If you use multiple regression to predict the scores from the two dummy
variables representing group membership, you obtain R2 = .40. Calculating f from the 2 ,
2
f 
.40
 .8165 . G*Power will convert 2 to f for you. Click Determine, Effect Size from Variance.
1  .40
Enter 2 and (1- 2). Click Calculate.
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The magnitude of Cohen’s f is affected by way in which the cases are allocated to the
populations. As shown above, when we had equal sample sizes f = .8165. If we changed the sample
sizes to 50 in each group, the f would remain the same (but power would be increased). Now look
what happens if we deviate from equal population sizes:
With most of
the scores in the
groups that differ most
from each other, f
increases.
With most of
the scores in the
group which differs
least from the others, f
drops.
Notes
Despite d, f, and his other effect size parameters being parameters, Cohen did not represent
them with Greek letters. Instead, he used bold-faced Roman characters.
Upon my request, Jacob Cohen (Jake) sent me several reprints of his articles. Every time the
reprint had a handwritten salutation and signature. I regret that I never met him in person.
My friend, and Institutional Research statistician, Chuck Rich, retired recently and passed on to
me Cohen’s book on power analysis. I always wanted that book, but always was too cheap to buy it
(I did buy his book on multiple regression). I have been enjoying reading Cohen’s own words. Thank
you, Chuck.
Karl L. Wuensch, January, 2016