Statistical Methods for Psychology

IN CHAPTER1 a distinction was drawn between measurement data (sometimes called quan-

titative data) and categorical data (sometimes called frequency data). When we deal with

measurement data, each observation represents a score along some continuum, and the

most common statistics are the mean and the standard deviation. When we deal with cate-

gorical data, on the other hand, the data consist of the frequencies of observations that fall

into each of two or more categories (e.g., “How many people rate their mom as their best

friend? ”).

In Chapter 5 we examined the use of the binomial distribution to test simple hypothe-

ses. In those cases, we were limited to situations in which an individual event had one of

only two possible outcomes, and we merely asked whether, over repeated trials, one out-

come occurred (statistically) significantly more often than the other. We will shortly see

how we can ask the same question using the chi-square test.

In this chapter we will expand the kinds of situations that we can evaluate. We will deal

with the case in which a single event can have two or morepossible outcomes, and then

with the case in which we have two independent variables and we want to test null hy-

potheses concerning their independence. For both of these situations, the appropriate sta-

tistical test will be the chi-square ( ) test.

The term chi-square (x^2 )has two distinct meanings in statistics, a fact that leads to

some confusion. In one meaning, it is used to refer to a particular mathematical distribu-

tion that exists in and of itself without any necessary referent in the outside world. In the

second meaning, it is used to refer to a statistical test that has a resulting test statistic dis-

tributed in approximately the same way as the distribution. When you hear someone re-

fer to chi-square, they usually have this second meaning in mind. (The test itself was

developed by Karl Pearson [1900] and is often referred to as Pearson’s chi-squareto dis-

tinguish it from other tests that also produce a statistic—for example, Friedman’s test,

discussed in Chapter 18, and the likelihood ratio tests discussed at the end of this chapter

and in Chapter 17.) You need to be familiar with both meanings of the term, however, if

you are to use the test correctly and intelligently, and if you are to understand many of the

other statistical procedures that follow.

6.1 The Chi-Square Distribution

The chi-square(x^2 )distributionis the distribution defined by

This is a rather messy-looking function and most readers will be pleased to know that they

will not have to work with it in any arithmetic sense. We do need to consider some of its fea-

tures, however, to understand what the distribution of is all about. The first thing that

should be mentioned, if only in the interest of satisfying healthy curiosity, is that the term

in the denominator, called a gamma function,is related to what we normally mean

by factorial. In fact, when the argument of gamma (k/2) is an integer, then

. We need gamma functions in part because arguments are not always
integers. Mathematical statisticians have a lot to say about gamma, but we’ll stop here.
A second and more important feature of this equation is that the distribution has only
one parameter (k). Everything else is either a constant or else the value of for which
we want to find the ordinate [ ]. Whereas the normal distribution was a two-parameter
function, with μand sas parameters, is a one-parameter function with kas the only
parameter. When we move from the mathematical to the statistical world, kwill become
our degrees of freedom. (We often signify the degrees of freedom by subscripting x^2.

x^2

f(x^2 )

x^2

≠(k/2)=[(k/2) 2 1]!

≠(k/2)

x^2

f(x^2 )=

1

2

k

(^2) ≠(k>2)
x2[(k>2)^2 1]e

• (X 22 )

x^2

x^2

x^2

140 Chapter 6 Categorical Data and Chi-Square

chi-square (x^2 )

Pearson’s
chi-square

gamma function

chi-square (x^2 )
distribution