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276 Part IV: Quality Assurance
respectively. For more details on binomial distribution, please refer to Gupta and
Walker (2007a).
Control Limits for the p Chart
To develop a p control chart proceed as follows:
1. From the process under investigation select m (m 25) samples of
size n (n 50) units. Note, however, if we have some prior information
or any clue that the process is producing a very small fraction of
nonconforming units, then the sample size should be large enough
so that the probability that it contains some nonconforming units is
relatively high.
2. Find the number of nonconforming units in each sample.
3. Find for each sample the fraction p i of nonconforming units, that is
x
p = (19.30)
i
n
where x is the number of nonconforming units in the ith (i = 1, 2,
. . . , m) sample.
4. Find the average nonconforming p – over the m samples, that is,
p + p + ... + p
1 2
p =
m
m
(19.31)
Part IV.B.4
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. Using the well-known result that the binomial distribution with
parameters n and p for large n can be approximated by the normal
distribution with mean np and variance np(1 – p), it can easily be seen
that p – will be approximately normally distributed with mean p and
standard deviation
p
( ) .
1 p
n
Hence, the upper and lower three-sigma control limits and the center
line for the p chart are as shown below.
UCL = p + 3
p( 1 p)
n
(19.32)
CL = p (19.33)