What About p-Charts?

When should we use the specialty charts for count data?

All charts for count-based data are charts for individual values. Regardless of whether we are working with a count or a rate, we obtain one value per time period and want to plot a point every time we get a value. This is why four specialty charts for count-based data had been developed before a general approach for charting individual values was discovered. These four charts are the p-chart, the np-chart, the c-chart, and the u-chart. The question addressed in this column is when to use these and other specialty charts with your count-based data.

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Deming and defective shoes

There is an interesting story in Quality Productivity and Competitive Position, page 208 - 209. Here is a percent defective for a process, and the X-mR chart would say it was in control at very tight limits. However, Dr Deming did look at the p-chart and noticed the pchart ucl and lcl were 4% and 15%, and the data ranged from 8% to 10%. This discrepancy led him to ask questions that determined the data were being falsified. It was thought that if the percent defective ever exceeded 10%, management would shut down the plant. Therefore, the head inspector made sure that the reported percent defective never exceeded 10%.

The vast majority of the data I work with are ESH&QA counts / counts per / and percentages. At least in that realm, I find the p, c, and u charts to be good "safety checks" for falsification or other manipulation of reporting.

Of course, I do agree that when in doubt, go xmR.

Thanks,

Reply to Steve

The story you relate did not come from Dr. Deming, it came from David S. Chambers and is found in Exercise 10.4 of our popular book, Understanding Statistical Process Control. Moreover, it was David that hit me over the head with the fact that all charts for count-based data are charts for individual values. This one instance does not justify the thousands of mistaken np charts and p charts that are created in error. As I said in the article, if you are sophisticated enough to know when the data ought to be binomial or Poisson, then you may use the specialty charts. But for everyone else using count-based data, the only safe approach is the XmR Chart.

(By the way, the inspector also kept the percentage above 8 percent in order to keep from loosing his job.)

Thanks

Thanks for the reply. Since I use the story in training, I can now attribute it to the correct source, and interesting about the 8% minimum. Non-statistical folks are not good at falsifying data . . .

I know Dr. Deming drew in stories and ideas from others, thought usually he was good at providing attribution to the source.

I do agree that there are a multitude of folks not applying the specialized charts properly.

XmR Control Limits Highly Variable

What I never see mentioned in the advocacy of XmR charts is that the control limits can be highly sensitive to the chronological order of the data. In Dr. Wheeler's Figure 4 Premum Freight Data example, the order of the data can make the XmR chart control limits as wide as 1.5 & 9.2, or as narrow as 4.2 & 6.5 (roughly the same as the p-chart limits in this case). To claim that the limits obtained for the particular order presented to be the "correct" limits ignores this variability. What if the various percentages had occurred in a different order? Similarly, in the Figure 3 example of On-time shipments, the data suffered particularly from autocorrelation, making the XmR chart limits much narrower than if the autocorrelation had been absent. If the order of the data had been different (reflecting less autocorrelation), the three points below the lower limits would then have been above the lower limits.

Enumerative vs Analytic

That is the whole point! The time order of data is what separates enumerative, or traditional statistics, from analytic methods.

Variability of the Opprtunity area

Dear Donald, thank you for this masterpiece. I have one questio about the area of opportunity in general.

If the area of opportunity varies a lot, how do I have to take into account this variability? For example, if in your example n.4 one month total shipments would be... 343, one order of magintude less than the others. Would the percentage of air freigfht shipments be significant as the others? Could I safely place it on the XmR chart together with other percentages? In my experience people tend to discard percentages which arise from areas of opportunity which are "very different" from the others.

Thank you for any advice!

Giuseppe

Order of Production

But the purpose of plotting data sequentially is to detect changes OVER TIME. You can't just arbitrary change the running order at whim to made a statistical point. A different time series would be a differnt process.

Rich DeRoeck

Back to the future

Some place in the archives of the DEN (Deming Electronic Network), probably in the late 90s, one will find that I made this very same point, and attempted to offer support via a comparison of using both a p chart and XmR on on simulated data. Random draws were taken a known poisson distribution but every now then a sample was taken from a different distribution (simulating a special or assignable cause). The test was would an XmR find the special cause as well as a p chart? Answer - yes. My follow up question back then was essentially why even bother teaching the p chart unless there is certainty that there is a circumstance that will ensure it is more robust and useful than using an XmR.

Indeed, it would be preferable for anyone using the Read Bead experiment to teach about variation to stop using np or p charts and only use an XmR.