Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

378 CHAPTER 12 | Introduction to Analysis of Variance
ANOVA typically involves a large number of scores and the mean is often not a
whole number. Therefore, it is usually much easier to calculate SS total
using the
computational formula:
SS 5SX 2 2 sSXd2
N
To make this formula consistent with the ANOVA notation, we substitute the letter
G in place of ΣX and obtain
SS total
5SX 2 2 G2
N
(12.3)
Applying this formula to the set of data in Table 12.2, we obtain
SS total
5 106 2 302
15
= 106 – 60
= 46
2. Within-Treatments Sum of Squares, SS within treatments
. Now we are looking at the
variability inside each of the treatment conditions. We already have computed the
SS within each of the three treatment conditions (Table 12.2): SS 1
= 6, SS 2
= 6,
and SS 3
= 4. To find the overall within-treatment sum of squares, we simply add
these values together:
SS within treatments
= ΣSS inside each treatment
(12.4)
For the data in Table 12.2, this formula gives
SS within treatments
= 6 + 6 + 4
= 16
3. Between-Treatments Sum of Squares, SS between treatments
. Before we introduce any
equations for SS between treatments
, consider what we have found so far. The total variability
for the data in Table 12.2 is SS total
= 46. We intend to partition this total into two
parts (see Figure 12.5). One part, SS within treatments
, has been found to be equal to 16.
This means that SS between treatments
must be equal to 30 so that the two parts (16 and 30)
to add up to the total (46). Thus, the value for SS between treatments
can be found simply
by subtraction:
To simplify the notation
we will use the
subscripts between
and within in place of
between treatments and
within treatments.
SS between
= SS total
– SS within
(12.5)
However, it is also possible to compute SS between
independently, using one of the two formulas
presented in Box 12.1. The advantage of computing all three SS values independently
is that you can check your calculations by ensuring that the two components, between and
within, add up to the total.