1. A problem with analysis you would like to have.

2. What happens when defect rates get very low? For Poisson data, the distribution gets very skewed and the usual three sigma limits are not valid. This occurs when λ≤20 and becomes very noticeable when λ is close to zero. This also happens for Binomial data when p is small and the sample size n is large. In this case the Binomial is approximately Poisson (n*p*(1-p)≥10 usually means it is a good approximation). For λ≤20 we can use table values to obtain control chart limits.

3. Table values from Edition 2 of the text:

4. Problem solved? Almost. The potential problem (which every SPC text ignores), is that when λ≤.1, if λ is small enough, the UCL should be such that any value X≥1 should be evidence of an out of control point. This violates the Shewhart principle that the ARL, or time between false signals, should be approximately equal to 100.

5. Is violating this Shewhart principle really a problem here? According to Dr. Tom, “ 沒問題 ”. Why? According to Shewhart, the ARL of 100 (or false signal rate of about 1%), was designed to keep the Operator from reacting to false signals. With Attributes charts, at very low defect rates the concepts of Special Cause and Defects overlap.

6. Is there another way to approach this without violating that Shewhart principle? Sort of. In Edition 2 of the text, the author suggests increasing the Area of Opportunity so that the ARL is still approximately 100, i.e. the chance of a false signal remains about 1%. The idea is that the chart may give false signals but retains “sensitivity”.

7. Is there any problem with increasing the Area of Opportunity here? According to Dr. Tom “ 很大問題 ”. If you increase the Area of Opportunity this way, it may take a number of defects over a long period of time in order to trigger a signal of Special Cause. By waiting it may make it more difficult to determine the source of the problem because time may have elapsed since one or more of the defects occurred, making it more difficult to determine what was “special” about the conditions when they happened.

8. So with defect rates that low one should? Look for the cause as soon as one knows about it. With low defect rates it is usually harder to find the cause, since defects are so rare there is not likely to be enough data to find a pattern in the conditions which produced them. A designed experiment may be the best and most economical way to reduce defects at this point. This is a problem you would like to have.