Applied Statistics and Probability for Engineers

6-1

6-8 MORE ABOUT PROBABILITY PLOTTING (CD ONLY)

Probability plots are extremely useful and are often the first technique used in an effort to

determine which probability distribution is likely to provide a reasonable model for the data.

We give a simple illustration of how a normal probability plot can be useful in distin-

guishing between normal and nonnormal data. Table S6-1 contains 50 observations gener-

ated at random from an exponential distribution with mean 20 (or ). These data

were generated using the random number generation capability in Minitab. Figure S6-1

presents a normal probability plot of these data, constructed using Minitab. The observa-

tions do not even approximately lie along a straight line, giving a clear indication that the

data do not follow a normal distribution. The strong curvature at both ends of the plot sug-

gests that the data come from a distribution with right or positive skew. Compare Fig. S6-1

with Fig. 6-19c.

Minitab also provides estimates of the mean and standard deviation of the distribution us-

ing the method of maximum likelihood(abbreviated ML on the graph in Figure S6-1). We

will discuss maximum likelihood estimation in Chapter 7. For the normal distribution, this is

the familiar sample mean and sample standard deviation that we first presented in Chapter 1.

Minitab also presents a quantitative measure of how well the data are described by a normal

distribution. This goodness-of-fit measure is called the Anderson-Darling statistic (abbrevi-

ated AD on the Minitab probability plot). The Anderson-Darling statistic is based on the prob-

ability integral transformation

that can be used to convert the data to a uniform distribution if the hypothesized distribution

is correct. Thus, if are independent and identically distributed random variables

whose cumulative distribution function is F(x), then are independent

uniform (0, 1) random variables. The Anderson-Darling statistic essentially compares how

close the F 1 x 12 , F 1 x 22 ,... , F 1 xn 2 values are to values from a uniform (0, 1) distribution. For

F 1 x 12 , F 1 x 22 ,... , F 1 xn 2

x 1 , x 2 ,... , xn

F 1 x 2 

x



f^1 u^2 du

0.05

0

0 50 100

5

10

20

30

40

50

60

70

80

90

95

99

Percentage

Data

Normal probability plot

ML estimates

ML estimates

Mean 20.7362

St. Dev. 19.2616

Goodness of fit

AD* 1.904

Figure S6-1. Normal

probability plot (from

Minitab) of the data

from Table S6-1.

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