Statistical Methods for Psychology

Thus, is read “chi-square with three degrees of freedom.” Alternatively, some authors

write it as .)

Figure 6.1 shows the plots for several different distributions, each representing a dif-

ferent value of k. From this figure it is obvious that the distribution changes markedly with

changes in k, becoming more symmetric as kincreases. It is also apparent that the mean

and variance of each distribution increase with increasing values of kand are directly

related to k. It can be shown that in all cases

6.2 The Chi-Square Goodness-of-Fit Test—One-Way Classification

We now turn to what is commonly referred to as the chi-square test,which is based on the

distribution. We will first examine the test as it is applied to one-dimensional tables and

then as applied to two-dimensional tables (contingency tables).

We will start with a simple but interesting example with only two categories and then

move on to an example with more than two categories. Our first example comes from a

paper on therapeutic touch that was published in the Journal of the American Medical As-

sociation (Rosa, Rosa, Sarner, and Barrett,1996). One of the things that made this an inter-

esting paper is that the second author, Emily Rosa, was only eleven years old at the time,

and she was the principal experimenter.^1 To quote from the abstract, “Therapeutic Touch (TT)

x^2

Variance= 2 k

Mean=k

x^2

x^2

x^2 (3)

x^23

Section 6.2 The Chi-Square Goodness-of-Fit Test—One-Way Classification 141

Density[f

(χ

2 )]

Density[f

(χ

2 )]

Density[f

(χ

2 )]

Density[f

(χ

2 )]

(a)df = 1

1 3 5 7 9 11 13 15

3.84

Chi-square (χ^2 ) Chi-square (χ^2 )

Chi-square (χ^2 ) Chi-square (χ^2 )

(c)df = 4

1 3 5 7 9 11 13 15

9.49

(b)df = 2

1 3 5 7 9 11 13 15

5.99

(d)df = 8

1 3 5 7 9 11 13 15 17 19

15.51

Figure 6.1 Chi-square distributions for df 5 1, 2, 4, and 8. (Arrows indicate critical

values at alpha 5 .05.)

(^1) The interesting feature of this paper is that Emily Rosa was an invited speaker at the “Ig Noble Prize” ceremony
sponsored by the Annals of Irreproducible Results,” located at MIT. This is a group of “whacky” scientists, to use a
psychological term, who look for and recognize interesting research studies. Ig Nobel Prizes honor “achievements
that cannot or should not be reproduced.” Emily’s invitation was meant as an honor, and true believers in therapeu-
tic touch were less than kind to her. The society’s web page is located at http://www.improb.com/ and I recommend
going to it when you need a break from this chapter.
chi-square test