Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

19.5 Normal Distribution 645

Probability

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

5 6 7 8 9 1011121314

Time (minutes)

The detailed shape of a normal-distribution curve is determined by its mean and standard

deviation values. For example, as shown in Figure 19.6, an experiment with a small standard

deviation will produce a tall, narrow curve; whereas a large standard deviation will result in a

short, wide curve. However, it is important to note that since the normal probability distribu-

tion represents all possible outcomes of an experiment (with the total of probabilities equal

to 1), the area under any given normal distribution should always be equal to 1. Also, note

normal distribution is symmetrical about the mean.

In statistics, it is customary and easier to normalize the mean and the standard deviation

values of an experiment and work with what is called thestandard normal distribution, which

has a mean value of zero ( 0) and a standard deviation value of 1 (s1). To do this, we

define what commonly is referred to as az scoreaccording to

(19.9)

In Equation (19.9),zrepresents the number of standard deviations from the mean. The mathe-

matical function that describes a normal-distribution curve or a standard normal curve is rather

complicated and may be beyond the level of your current understanding. Most of you will

learn about it later in your statistics or engineering classes. For now, using Excel, we have generated

a table that shows the areas under portions of the standard normal-distribution curve, shown in

Table 19.11. At this stage of your education, it is important for you to know how to use the table

and solve some problems. A more detailed explanation will be provided in your future classes. We

will next demonstrate how to use Table 19.11, using a number of example problems.

z

xx

s

x

Small standard

deviation

Mean

Large standard

deviation

Mean

■Figure 19.6

The shape of a normal

distribution curve as

determined by its mean and

standard deviation.

■Figure 19.5

Plot of probability distribution

for Example 19.4.

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