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

SECTION 12.3 | ANOVA Notation and Formulas 375
LEARNING CHECK
1. In an analysis of variance, the primary effect of large mean differences from one
sample to another is to increase the value for ______.
a. the variance between treatments.
b. the variance within treatments
c. the total variance.
d. large mean differences will not directly affect any of the three variances.
2. When the null hypothesis is true for an ANOVA, what is the expected value for the
F-ratio?
a. 0
b. 1.00
c. much greater than 1.00
d. less than zero
ANSWERS
1. A, 2. B
12.3 ANOVA Notation and Formulas
LEARNING OBJECTIVE
5. Calculate the three SS values, the three df values, and the two mean squares
(MS values) that are needed for the F-ratio and describe the relationships among
them.
Because ANOVA typically is used to examine data from more than two treatment conditions
(and more than two samples), we need a notational system to keep track of all the
individual scores and totals. To help introduce this notational system, we use the hypothetical
data from Table 12.1 again. The data are reproduced in Table 12.2 along with some of
the notation and statistics that will be described.
1. The letter k is used to identify the number of treatment conditions—that is, the
number of levels of the factor. For an independent-measures study, k also specifies
the number of separate samples. For the data in Table 12.2, there are three treatments,
so k = 3.
2. The number of scores in each treatment is identified by a lowercase letter n. For the
example in Table 12.2, n = 5 for all the treatments. If the samples are of different
sizes, you can identify a specific sample by using a subscript. For example, n 2
is
the number of scores in treatment 2.
3. The total number of scores in the entire study is specified by a capital letter N.
When all the samples are the same size (n is constant), N = kn. For the data in
Table 12.2, there are n = 5 scores in each of the k = 3 treatments, so we have a
total of N = 3(5) = 15 scores in the entire study.
4. The sum of the scores (ΣX) for each treatment condition is identified by the capital
letter T (for treatment total). The total for a specific treatment can be identified by
adding a numerical subscript to the T. For example, the total for the second treatment
in Table 12.2 is T 2
= 5.