Higher Engineering Mathematics

J

Statistics and probability

58

The normal distribution

58.1 Introduction to the normal

distribution

When data is obtained, it can frequently be consid-
ered to be a sample (i.e. a few members) drawn at
random from a large population (i.e. a set having
many members). If the sample number is large, it is
theoretically possible to choose class intervals which
are very small, but which still have a number of mem-
bers falling within each class. A frequency polygon
of this data then has a large number of small line
segments and approximates to a continuous curve.
Such a curve is called afrequency or a distribution
curve.
An extremely important symmetrical distribution
curve is called thenormal curveand is as shown
in Fig. 58.1. This curve can be described by a math-
ematical equation and is the basis of much of the
work done in more advanced statistics. Many natu-
ral 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 differ-

ent units and

(d) by having different areas between the curve and
the horizontal axis.

A normal distribution curve isstandardizedas

follows:

(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

horizontal axis is made equal to unity.

When a normal distribution curve has been stan-

dardized, 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

√

(2π)

e

(

z^2

2

)

dz, wherez=

x−x

σ

Probability

density

Standard deviations

z 1 0 z 2 z-value

Figure 58.2

To save repeatedly determining the values of

this function, tables of partial areas under the

standardized normal curve are available in many