Statistical Methods for Psychology

assumed) it doesn’t automatically suggest a reason for having an appendix that tells all

about it. The reason is quite simple. By using Appendixz, we can readily calculate the

probability that a score drawn at random from the population will have a value lying be-

tween any two specified points ( and ). Thus, by using the appropriate table we can

make probability statements in answer to a variety of questions. You will see examples of

such questions in the rest of this chapter. They will also appear in many other chapters

throughout the book.

3.2 The Standard Normal Distribution

A problem arises when we try to table the normal distribution, because the distribution de-

pends on the values of the mean and the standard deviation (mand ) of the distribution.

To do the job right, we would have to make up a different table for every possible combi-

nation of the values of mand s, which certainly is not practical. The solution to this prob-

lem is to work with what is called the standard normal distribution,which has a mean of 0

and a standard deviation of 1. Such a distribution is often designated as N(0,1), where N

refers to the fact that it is normal, 0 is the value of m, and 1 is the value of. (N(m, ) is

the more general expression.) Given the standard normal distribution in the appendix and a

set of rules for transforming any normal distribution to standard form and vice versa, we

can use Appendixzto find the areas under any normal distribution.

Consider the distribution shown in Figure 3.6, with a mean of 50 and a standard deviation

of 10 (variance of 100). It represents the distribution of an entire populationof Total Behavior

Problem scores from the Achenbach Youth Self-Report form, of which the data in Figures 3.3

and 3.4 are a sample. If we knew something about the areas under the curve in Figure 3.6, we

could say something about the probability of various values of Behavior Problem scores and

could identify, for example, those scores that are so high that they are obtained by only 5% or

10% of the population. You might wonder why we would want to do this, but it is often impor-

tant in diagnosis to be able to separate extreme scores from more typical scores.

The only tables of the normal distribution that are readily available are those of the

standardnormal distribution. Therefore, before we can answer questions about the proba-

bility that an individual will get a score above some particular value, we must first trans-

form the distribution in Figure 3.6 (or at least specific points along it) to a standard normal

distribution. That is, we want to be able to say that a score of from a normal distribution

with a mean of 50 and a variance of 100—often denoted N(50,100)—is comparable to a

Xi

s^2 s^2

s

X 1 X 2

Section 3.2 The Standard Normal Distribution 71

X: 20

f(

X

)

0.40

0.30

0.20

0.10

30 40 50 60 70 80

X– μ: –30 –20 –10 0 10 20 30

z:3–3 –2 –1 0 1 2

Figure 3.6 A normal distribution with various transformations on the abscissa

standard normal
distribution