Statistical Methods for Psychology

11.4 Calculations in the Analysis of Variance

At this point we will use the example from Eysenck to illustrate the calculations used in

the analysis of variance. Even though you may think that you will always use computer

software to run analyses of variance, it is very important to understand how you would

carry out the calculations using a calculator. First of all, it helps you to understand the

basic procedure. In addition, it makes it much easier to understand some of the controver-

sies and alternative analyses that are proposed. Finally, no computer program will do

everything you want it to do, and you must occasionally resort to direct calculations. So

bear with me on the calculations, even if you think that I am wasting my time.

Sum of Squares

In the analysis of variance much of our computation deals with sums of squares. As we

saw in Chapter 9, a sum of squares is merely the sum of the squared deviations about the

mean or, more often, some multiple of that. When we first defined the sam-

ple variance, we saw that

Here, the numerator is the sum of squaresof Xand the denominator is the degrees of free-

dom. Sums of squares have the advantage of being additive, whereas mean squares and vari-

ances are additive only if they happen to be based on the same number of degrees of freedom.

The Data

The data are reproduced in Table 11.2, along with a boxplot of the data in Figure 11.2 and

the calculations in Table 11.3. We will discuss the calculations and the results in detail.

Because these actual data points are fictitious (although the means and variances are not),

there is little to be gained by examining the distribution of observations within individual

s^2 X= a

(X 2 X)^2

n 21

=

aX

(^22) A
aXB
(^2) >n
n 21
Ca(X 2 X)^2 D
324 Chapter 11 Simple Analysis of Variance
Table 11.2 Data for example from Eysenck (1974)
Counting Rhyming Adjective Imagery Intentional Total
9 7 11 12 10
8 9 13 11 19
6681614
86611 5
10 6 14 9 10
4111123 11
6 6 13 12 14
5 3 13 10 15
7 8 10 19 11
7 7 11 11 11
Mean 7.00 6.90 11.00 13.40 12.00 10.06
St. Dev. 1.83 2.13 2.49 4.50 3.74 4.01
Variance 3.33 4.54 6.22 20.27 14.00 16.058
sums of squares