Statistical Methods for Psychology

76 Chapter 3 The Normal Distribution

the limits encompassing 95% of the population, we want to find those scores that are 1.96

standard deviations above and below the mean of the population. This can be written as

where the values of Xcorresponding to and represent the limits

we seek. For our example the limits will be

Limits 550 6 (1.96)(10) 550 6 19.6 5 30.4 and 69.6.

So the probability is .95 that a child’s score (X) chosen at random would be between

30.4 and 69.6. We may not be very interested in low scores, because they don’t represent

problems. But anyone with a score of 69.6 or higher is a problem to someone. Only 2.5%

of children score at least that high.

What we have just discussed is closely related to, but not quite the same as, what we

will later consider under the heading of confidence limits. The major difference is that here

we knew the population mean and were trying to estimate where a single observation (X)

would fall. When we discuss confidence limits, we will have a sample mean (or some other

statistic) and will want to set limits that have a probability of .95 of bracketing the popula-

tion mean (or some other relevant parameter). You do not need to know anything at all

about confidence limits at this point. I simply mention the issue to forestall any confusion

in the future.

3.5 Assessing Whether Data Are Normally Distributed

There will be many occasions in this book where we will assume that data are normally

distributed, but it is difficult to look at a distribution of sample data and assess the reason-

ableness of such an assumption. Statistics texts are filled with examples of distributions

(m11.96s) (m21.96s)

X=m61.96s

X2m= 6 1.96s

6 1.96=

X2m

s

z=

X2m

s

f(

X

)

0.40

0.30

0.20

0.10

0

95%

–3.0

z

–2.0 –1.0 0 1.0 2.0 3.0

Figure 3.9 Values of zthat enclose 95% of the behavior problem scores