Statistical Methods for Psychology

Although our player should throw each symbol equally often, our data suggest that she

is throwing Rock more often than would be expected. However this may just be a random

deviation due to chance. Even if you are deliberately randomizing your throws, one is

likely to come out more frequently than others. (Moreover, people are notoriously poor at

generating random sequences.) What we want is a goodness-of-fit test to ask whether the

deviations from what would be expected by chance are large enough to lead us to conclude

that this child’s throws weren’t random, but that she was really throwing Rock at greater

than chance levels.

The statistic for these data using the observed and expected frequencies given in

Table 6.3 follows. Notice that it is a simple extension of what we did when we had two

categories.

In this example we have three categories and thus 2 df. The critical value of on 2 df 5

5.99, and we have no reason to doubt that our player was equally likely to throw each

symbol.

6.3 Two Classification Variables: Contingency Table Analysis

In the previous examples we considered the case in which data are categorized along only

one dimension (classification variable). More often, however, data are categorized with

respect to two (or more) variables, and we are interested in asking whether those variables

are independent of one another. To put this in the reverse, we often are interested in asking

whether the distribution of one variable is contingenton a second variable. (Statisticians

often use the phrase “conditional on” instead of “contingent on,” but they mean the same

thing. I mention this because you will see the word “conditional” appearing often in this

chapter.) In this situation we will construct a contingency tableshowing the distribution

of one variable at each level of the other variable. A good example of such a test concerns

the controversial question of whether or not there is racial bias in the assignment of death

sentences.

There have been a number of studies over the years looking at whether the imposition

of a death sentence is affected by the race of the defendant (and/or the race of the victim).

You will see an extended example of such data in Exercise 6.41. Peterson (2001) reports

data on a study by Unah and Borger (2001) examining the death penalty in North Carolina

in 1993–1997. The data in Table 6.4 show the outcome of sentencing for white and non-

white (mostly black and Hispanic) defendants when the victim was white. The expected

frequencies are shown in parentheses.

Expected Frequencies for Contingency Tables

The expected frequencies in a contingency table represent those frequencies that we would

expect if the two variables forming the table (here, race and sentence) were independent.

For a contingency table the expected frequency for a given cellis obtained by multiplying

x^2

=1.68

=

(30–25)^2

25

1

(21–25)^2

25

1

(24–25)^2

25

=

52142112

25

x^2 = a

(O 2 E)^2

E

x^2

Section 6.3 Two Classification Variables: Contingency Table Analysis 145

contingency
table

cell