Supplemental Material for Chapter 7
S7.1. Probability Limits on Control Charts
In Chapters 6 and 7 of the textbook, probability limits for control charts are briefly discussed. The usual three-sigma limits are almost always used with variables control charts, although as we point out, there can be some occasional advantage to the use of probability limits, such as on the range chart to obtain a non-zero lower control limit.
The standard applications of attributes control charts almost always use the three-sigma limits as well, although their use is potentially somewhat more troublesome here. When three-sigma limits are used on attributes charts, we are basically assuming that the normal approximation to either the binomial or Poisson distribution is appropriate, at least to the extent that the distribution of the attribute chart statistic is approximately symmetric, and that the symmetric three-sigma control limits are satisfactory.
This will, of course, not always be the case. If the binomial probability p is small and the sample size n is not large, or if the Poisson mean is small, then symmetric three-sigma control limits on the p or c chart may not be appropriate, and probability limits may be much better.
For example, consider a p chart with p = 0.07 and n = 100. The center line is at 0.07 and the usual three-sigma control limits are UCL = -0.007 = 0 and UCL = 0.147. A short table of cumulative binomial probabilities computed from Minitab follows.
Cumulative Distribution Function
Binomial with n = 100 and p = 0.0700000
x P( X <= x )
0.00 0.0007
1.00 0.0060
2.00 0.0258
3.00 0.0744
4.00 0.1632
5.00 0.2914
6.00 0.4443
7.00 0.5988
8.00 0.7340
9.00 0.8380
10.00 0.9092
11.00 0.9531
12.00 0.9776
13.00 0.9901
14.00 0.9959
15.00 0.9984
16.00 0.9994
17.00 0.9998
18.00 0.9999
19.00 1.0000
20.00 1.0000
If the lower control limit is at zero and the upper control limit is at 0.147, then any sample with 15 or more defective items will plot beyond the upper control limit. The above table shows that the probability of obtaining 15 or more defectives when the process is in-control is 1 – 0.9959 = 0.0041. This is about 50% greater than the false alarm rate on the normal-theory three-sigma limit control chart (0.0027). However, if we were to set the lower control limit at 0.01 and the upper control limit at 0.15, and conclude that the process is out-of-control only if a control limit is exceeded, than the false alarm rate is 0.0007 + 0.0016 = 0.0023, which is very close to the advertised value of 0.0027. Furthermore, there is a nonzero LCL, which can be very useful in practice. Notice that the control limits are not symmetric around the center line. However, the distribution of is not symmetric, so this should not be too surprising.
There are several other interesting approaches to setting probability-type limits on attribute control charts. Refer to Ryan (2000), Acosta-Mejia (1999), Ryan and Schwertman (1997), Schwertman and Ryan (1997), and Shore (2000).
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