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Chapter 19: B. Statistical Process Control 281
x
p = (19.35)
i
n
where x is the number of nonconforming units in the ith (i = 1, 2,
. . ., m) sample
4. Find the average fraction p – of nonconforming units over the m
samples, that is
np + np + … + n p
1 1 2 2
m
p =
n + n + n + … + n
1 2 3
i
m
m
. (19.36)
The value of p – determines the center line for the p chart and is
an estimate of p, the process fraction of nonconforming units.
5. The control limits for p with variable sample sizes are determined
for each sample separately. Thus, for example, the upper and lower
three-sigma control limits for the ith sample are
UCL = p + 3
p( 1 p)
n i
(19.37)
CL = p (19.38)
LCL = p 3
p( 1 p)
. (19.39)
n i
Note that the center line is the same for all samples whereas the control limits will
be different for different samples.
To illustrate the construction of the p chart with variable sample size we
consider the data in Table 19.6 of Example 19.7.
Part IV.B.4
EXAMPLE 19.7
Suppose, in Example 19.6, that all the chips manufactured during a certain fixed period
of time are inspected each day. However, the number of computer chips manufactured
varies during that fixed period each day. The data collected for the study period
of 30 days is as shown in Table 19.6. Construct the p chart for these data and determine
whether the process is stable or not.
Solution:
Using the data in Table 19.6 and equations (19.37) through (19.39), we get the trial
control limits
UCL = 0.01675, CL = 0.00813, LCL = 0.0.
Continued