Statistical Methods for Psychology

interested in the distribution of errors in astronomical observations. In fact, the normal

distribution is variously referred to as the Gaussian distribution and as the “normal law of

error.” Adolph Quetelet (1796–1874), a Belgian astronomer, was the first to apply the dis-

tribution to social and biological data. Apparently having nothing better to do with his time,

he collected chest measurements of Scottish soldiers and heights of French soldiers. He

found that both sets of measurements were approximately normally distributed. Quetelet

interpreted the data to indicate that the mean of this distribution was the ideal at which na-

ture was aiming, and observations to each side of the mean represented error (a deviation

from nature’s ideal). (For males like myself, it is somehow comforting to think of all

those bigger guys as nature’s mistakes.) Although we no longer think of the mean as

nature’s ideal, this is a useful way to conceptualize variability around the mean. In fact, we

still use the word errorto refer to deviations from the mean. Francis Galton (1822–1911)

carried Quetelet’s ideas further and gave the normal distribution a central role in psycho-

logical theory, especially the theory of mental abilities. Some would insist that Galton was

toosuccessful in this endeavor, and we tend to assume that measures are normally distrib-

uted even when they are not. I won’t argue the issue here.

Mathematically the normal distribution is defined as

wherepand eare constants (p53.1416 and e 5 2.7183), and mand sare the mean and

the standard deviation, respectively, of the distribution. If mand sare known, the ordinate,

f(X), for any value of Xcan be obtained simply by substituting the appropriate values for

m, s, and Xand solving the equation. This is not nearly as difficult as it looks, but in prac-

tice you are unlikely ever to have to make the calculations. The cumulative form of this dis-

tribution is tabled, and we can simply read the information we need from the table.

Those of you who have had a course in calculus may recognize that the area under the

curve between any two values of X(say and ), and thus the probability that a ran-

domly drawn score will fall within that interval, can be found by integrating the function

over the range from to. Those of you who have not had such a course can take com-

fort from the fact that tables are readily available in which this work has already been done

for us or by use of which we can easily do the work ourselves. Such a table appears in

Appendixz (p. 720).

You might be excused at this point for wondering why anyone would want to table such

a distribution in the first place. Just because a distribution is common (or at least commonly

X 1 X 2

X 1 X 2

f(X)=

1

s 22 p

(e)^2 (X2m)

(^2) /2s 2

5 ¿ 8 –

70 Chapter 3 The Normal Distribution

X

f(

X

) (density)

0.40

0.30

0.20

0.10

0.05

0

–3 –2 –1 012 3 4

0.15

0.25

0.35

–4

Figure 3.5 A characteristic normal distribution with values of Xon the abscissa and

density on the ordinate