Statistical Methods for Psychology

considerations ( , , and s), about which we can do relatively little, from sample char-

acteristics (n), over which we have more control. Although this produces no basic change

in the underlying theory, it makes the concept easier to understand and use.

8.4 Power Calculations for Differences Between Two Independent Means

When we wish to test the difference between two independent means, the treatment of power

is very similar to our treatment of the case that we used for only one mean. In Section 8.3 we

obtained dby taking the difference between munder and munder and dividing by s. In

testing the difference between two independent means, we will do basically the same thing,

although this time we will work with mean differences. Thus, we want the difference between

the two population means ( ) under minus the difference ( ) under ,

divided by s. (Recall that we assume .) In all usual applications, however,

( ) under is zero, so we can drop that term from our formula. Thus,

where the numerator refers to the difference to be expected under and the denominator

represents the standard deviation of the populations. You should recognize that this is the

same dthat we saw in Chapter 7 where it was also labeled Cohen’s d, or sometimes

Hedges g. The only difference is that here it is expressed in terms of population means

rather than sample means.

In the case of two samples, we must distinguish between experiments involving equal

ns and those involving unequal ns. We will treat these two cases separately.

Equal Sample Sizes

Assume we wish to test the difference between two treatments and either expect that the

difference in population means will be approximately 5 points or else are interested only in

finding a difference of at least 5 points. Further assume that from past data we think that s

is approximately 10. Then

Thus, we are expecting a difference of one-half of a standard deviation between the two

means, what Cohen (1988) would call a moderate effect.

First we will investigate the power of an experiment with 25 observations in each of

two groups. We will define din the two-sample case as

where n 5 the number of cases in any one sample(there are 2ncases in all). Thus,

From Appendix Power, by interpolation for d51.77 with a two-tailed test at a5.05,

power 5 .43. Thus, if our investigator actually runs this experiment with 25 subjects,

=1.77

d=(0.50)

A

25

2

=0.50 1 12.5=0.50(3.54)

d=d

A

n

2

d=

m 1 2m 2

s

=

5

10

=0.50

H 1

d=

(m 1 2m 2 ) 2 (0)

s

=

m 1 2m 2

s

m 1 2m 2 H 0

s^21 =s^22 =s^2

m 1 2m 2 H 1 m 1 2m 2 H 0

H 1 H 0

m 0 m 1

Section 8.4 Power Calculations for Differences Between Two Independent Means 233