Statistical Methods for Psychology

variance of the population from which the scores were drawn. Because we have assumed

that all populations have the same variance, it is also one estimate of the common popula-

tion variance. If you prefer, you can think of

,,,

where is read as “is estimated by.” Because of our homogeneity assumption, all these

are estimates of. For the sake of increased reliability, we can pool the five estimates by

taking their mean, if , and thus

where k 5 the number of treatments (in this case, five).^1 This gives us one estimate of the

population variance that we will later refer to as MSerror(read “mean square error”), or,

sometimes, MSwithin.It is important to note that this estimate does not depend on the truth

or falsity of , because is calculated on each sample separately. For the data from

Eysenck’s study, our pooled estimate of will be

Now let us assume that is true. If this is the case, then our five samples of 10 cases

can be thought of as five independent samples from the same population (or, equivalently,

from five identical populations), and we can produce another possible estimate of. Re-

call from Chapter 7 that the central limit theorem states that the variance of means drawn

from the same population equals the variance of the population divided by the sample size.

If is true, the sample means have been drawn from the same population (or identical

ones, which amounts to the same thing), and therefore the variance of our five sample

means estimates.

where nis the size of each sample. Thus, we can reverse the usual order of things and cal-

culate the variance of our sample means ( ) to obtain the second estimate of :

This term is referred to as MStreatmentoften abbreviated as ; we will return to it shortly.

We now have two estimates of the population variance ( ). One of these estimates

( ) is independent of the truth or falsity of. The other ( ) is an estimate

of only as long as is true (only as long as the conditions of the central limit theorem

are met; namely, that the means are drawn from one population or several identical popula-

tions). Thus, if the two estimates agree, we will have support for the truth of , and if they

disagree, we will have support for the falsity of.^2

From the preceding discussion, we can concisely state the logic of the analysis of

variance. To test , we calculate two estimates of the population variance—one that is

independent of the truth or falsity of H 0 , and another that is dependent on H 0. If the two

H 0

H 0

H 0

s^2 e H 0

MSerror H 0 MStreatment

s^2 e

MStreat

s^2 ensX^2

sX^2 s^2 e

s^2 e

n

sX^2

s^2 e>n

H 0

s^2 e

H 0

s^2 e(3.33 1 4.54 1 6.22 1 20.27 1 14.00)> 5 =9.67

s^2 e

H 0 s^2 j

s^2 es^2 es^2 jas^2 j>k

n 1 =n 2 = Á =n 5

s^2 e



s 12 s 12 s^22 s^22 Á s^2 es^2 e

s^2 e

322 Chapter 11 Simple Analysis of Variance

(^1) If the sample sizes were not equal, we would still average the five estimates, but in this case we would weight
each estimate by the number of degrees of freedom for each sample—just as we did in Chapter 7.
(^2) Students often have trouble with the statement that “means are drawn from the same population” when we know
in fact that they are often drawn from logically distinct populations. It seems silly to speak of means of males and
females as coming from one population when we know that these are really two different populations of people.
However, if the population of scores for females is exactly the same as the population of scores for males, then we
can legitimately speak of these as being the identical (or the same) population of scores, and we can behave
accordingly.
MSerror
MSwithin
MStreatment