Chapter 58

The normal distribution

58.1 Introduction to the normal

distribution

When data is obtained, it can frequently be considered
to be a sample (i.e. a few members) drawn at random
from a large population (i.e. a set having many mem-
bers). If the sample number is large, it is theoretically
possible to choose class intervals which are very small,
but which stillhave a number of members fallingwithin
each class. A frequency polygon of this data then has a
large number of small line segments and approximates
toacontinuouscurve.Suchacurveiscalledafrequency
or a distribution curve.
An extremely important symmetrical distributioncurve
is called thenormal curveand is as shown in Fig. 58.1.
This curve can be described by a mathematical equa-
tion and is the basis of much of the work done in more
advanced statistics. Many natural occurrences such as
the heights or weights of a group of people, the sizes
of components produced by a particular machine and
the life length of certain components approximate to a
normal distribution.

Variable

Frequency

Figure 58.1

Normal distribution curves can differ from one another
in the following four ways:

(a) by having different mean values

(b) by having different values of standard deviations

(c) the variables having different values and different

units and

(d) by having different areas between the curve and

the horizontal axis.

A normal distribution curve isstandardizedas fol-

lows:

(a) The mean value of the unstandardized curve is

made the origin, thus making the mean value,

x, zero.

(b) The horizontal axis is scaled in standard devia-

tions. This is done by lettingz=

x−x

σ

,where

zis called thenormal standard variate,xis the

value of the variable,xis the mean value of the

distribution andσis the standard deviation of the

distribution.

(c) The area between the normal curve and the hori-

zontal axis is made equal to unity.

When a normal distribution curve has been standard-

ized, the normal curve is called astandardized normal

curveor anormal probability curve, and any normally

distributed data may be represented by thesamenormal

probability curve.

The area under part of a normal probability curve is

directly proportional to probability and the value of the

shaded area shown in Fig. 58.2 can be determined by

evaluating:

∫

1

√

( 2 π)

e

(

z^2

2

)

dz, wherez=

x−x

σ

To save repeatedly determining the values of this func-

tion, tables of partial areas under the standardized nor-

mal curve are available in many mathematical formulae

books, and such a table is shown in Table 58.1, on

page 564.