Statistical Methods for Psychology

SStotal

The SStotal(read “sum of squares total”) represents the sum of squares of all the observa-

tions, regardless of which treatment produced them. Letting represent the grand mean,

the definitional formula is

This is a term we saw much earlier when we were calculating the variance of a set of

numbers, and is the numerator for the variance. (The denominator was the degrees of free-

dom.) This formula, like the ones that follow, is probably not the formula we would use if

we were to do the hand calculations for this problem. The formulae are very susceptible to

the effects of rounding error. However, they are perfectly correct formulae, and represent

the way that we normally think about the analysis. For those who prefer more traditional

hand-calculation formulae, they can be found in earlier editions of this book.

SStreat

The definitional formula for SStreatis framed in the context of deviations of group means

from the grand mean. Here we have

You can see that SStreatis just the sum of squared deviations of the treatment means around

the grand mean, multiplied by nlater to give us an estimate of the population variance.

SSerror

In practice, SSerroris obtained by subtraction. Since it can be easily shown that

then it must also be true that

This is the procedure presented in Table 11.3, and it makes our calculations easier.

To present SSerrorin terms of deviations from means, we can write

Here you can see that SSerroris simply the sum over groups of the sums of squared de-

viation of scores around their group’s mean. This approach is illustrated in the following,

where I have calculated the sum of squares within each of the groups. Notice that for each

group there is absolutely no influence of data from other groups, and therefore the truth or

falsity of the null hypothesis is irrelevant to the calculations.

SSerror= 435.30

SSwithin International=a 1 (10 2 12.00)^21 (19 2 12.00)^21... 1 (11 2 12.00)^22 =126.00

SSwithin Imagery=a 1 (12 2 13.4)^21 (11 2 13.4)^21... 1 (11 2 13.4)^22 =182.40

SSwithin Adjective=a 1 (11 2 11.00)^21 (13 2 11.00)^21... 1 (11 2 11.00)^22 = 56.00

SSwithin Rhyming=a 1 (7 2 6.90)^21 (9 2 6.90)^21... 1 (7 2 6.90)^22 = 40.90

SSwithin Counting=a 1 (9 2 7.00)^21 (8 2 7.00)^21... 1 (7 2 7.00)^22 = 30.00

SSerror= a(Xij 2 Xj)^2

SSerror=SStotal 2 SStreat

SStotal=SStreat 1 SSerror

SStreat=na(Xj 2 X..)^2

SStotal= a(Xij 2 X..)^2

X..

326 Chapter 11 Simple Analysis of Variance

SStotal

SStreat

SSerror