Statistical Methods for Psychology

When we sum these individual terms, we obtain 435.30, which agrees with the answer we

obtained in Table 11.3.

The Summary Table

Table 11.3 also shows the summary tablefor the analysis of variance. It is called a

summary table for the rather obvious reason that it summarizes a series of calculations,

making it possible to tell at a glance what the data have to offer. In older journals you

will often find the complete summary table displayed. More recently, primarily to save

space, usually just the resulting Fs (to be defined) and the degrees of freedom are

presented.

Sources of Variation

The first column of the summary table contains the sources of variation—the word “variation”

being synonymous with the phrase “sum of squares.” As can be seen from the table, there are

three sources of variation: the variation due to treatments (variation among treatment means),

the variation due to error (variation within the treatments), and the total variation. These

sources reflect the fact that we have partitioned the total sum of squares into two portions, one

representing variability within the individual groups and the other representing variability

among the several group means.

Degrees of Freedom

The degrees of freedom column in Table 11.3 represents the allocation of the total

number of degrees of freedom between the two sources of variation. With 49 dfoverall

(i.e., N 2 1), four of these are associated with differences among treatment means and

the remaining 45 are associated with variability within the treatment groups. The cal-

culation of dfis probably the easiest part of our task. The total number of degrees of

freedom (dftotal)is always N 2 1, where Nis the total number of observations. The num-

ber of degrees of freedom between treatments (dftreat)is always k 2 1, where kis the

number of treatments. The number of degrees of freedom for error (dferror)is most eas-

ily thought of as what is left over and is obtained by subtracting from.

However, dferrorcan be calculated more directly as the sum of the degrees of freedom

within each treatment.

To put this in a slightly different form, the total variability is based on Nscores and there-

fore has N 2 1 df. The variability of treatment means is based on kmeans and therefore has

k 2 1 df. The variability within any one treatment is based on nscores, and thus has n 2 1 df,

but since we sum kof these within-treatment terms, we will have ktimes n 2 1 or k(n 2 1) df.

Mean Squares

We will now go to the MScolumn in Table 11.3. (There is little to be said about

the column labeled SS; it simply contains the sums of squares obtained in the section

on calculations.) The column of mean squares contains our two estimates of. These

values are obtained by dividing the sums of squares by their corresponding df. Thus,

351.52/4 5 87.88 and 435.30/45 5 9.67. We typically do not calculate , because

we have no need for it. If we were to do so, this term would equal 786.82/49 5 16.058,

which, as you can see from Table 11.3, is the variance of all Nobservations, regardless

of treatment. Although it is true that mean squares are variance estimates, it is impor-

tant to keep in mind what variances these terms are estimating. Thus, MSerroris an

MStotal

s^2 e

dftreat dftotal

Section 11.4 Calculations in the Analysis of Variance 327

summary table

dferror

dftreat

dftotal