Applied Statistics and Probability for Engineers

626 CHAPTER 16 STATISTICAL QUALITY CONTROL

These control limits are based on the normal approximation to the binomial distribu-

tion. When pis small, the normal approximation may not always be adequate. In such

cases, we may use control limits obtained directly from a table of binomial probabilities.

If is small, the lower control limit obtained from the normal approximation may be a

negative number. If this should occur, it is customary to consider zero as the lower control

limit.

EXAMPLE 16-4 Suppose we wish to construct a fraction-defective control chart for a ceramic substrate pro-

duction line. We have 20 preliminary samples, each of size 100; the number of defectives in

each sample is shown in Table 16-4. Assume that the samples are numbered in the sequence

of production. Note that (8002000)  0.40; therefore, the trial parameters for the con-

trol chart are

The control chart is shown in Fig. 16-16. All samples are in control. If they were not, we

would search for assignable causes of variation and revise the limits accordingly. This chart

can be used for controlling future production.

Although this process exhibits statistical control, its defective rate ( ) is very

poor. We should take appropriate steps to investigate the process to determine why such a

large number of defective units is being produced. Defective units should be analyzed to de-

termine the specific types of defects present. Once the defect types are known, process

changes should be investigated to determine their impact on defect levels. Designed experi-

ments may be useful in this regard.

Computer software also produces an NPchart.This is just a control chart of , the

number of defectives in a sample. The points, center line, and control limits for this chart are

just multiples (times n) of the corresponding elements of a Pchart. The use of an NPchart

avoids the fractions in a Pchart.

nPˆD

p0.40

LCL0.40 3

B

1 0.40 21 0.60 2

100

0.25

UCL0.40 3

B

1 0.40 21 0.60 2

100

0.55 CL0.40

p

p

Table 16-4 Number of Defectives in Samples of 100

Ceramic Substrates

Sample No. of Defectives Sample No. of Defectives

14411 36

24812 52

33213 35

45014 41

52915 42

63116 30

74617 46

85218 38

94419 26

10 48 20 30

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