Written by Rick Haynes on November 7, 2009 – 11:05 am
Part of the teachings of Forrest Breyfogle, and even his recent correspondence partner Donald Wheeler, is that the individuals chart (I-chart) is a better method to monitor a process defect rate than using a p-chart.
Yes, I know that we were all taught that a p-chart is the correct Shewhart chart to use for defect rates when we have an unequal (and equal) sample size. One of my mentors told me, in jest, that the p in a p-chart meant “probably should use an I-chart.” I always thought it was cute until I began examining my history of p-chart success. I found it was nearly always out of control, even when applied to processes that were producing the same average defect rate month to month over many years. You may have seen the same things.
I now teach people the unspoken assumptions to a p-chart and just like Breyfogle and Wheeler, recommend people avoid the p-chart. What we were never taught was that the p-chart control limits are are based on an ideal binomial distribution data population. If the data stream is not producing perfect binomial data, it will be out of control.
What is a perfect binomial, this would mean that every item or transaction has an identical probability of being defective. When is this ever true? Every place I have ever worked experiences defect causes that come and go based on interactions, product mix, raw material batches, personnel changes……….. Some days are better than others. But over time, the average defect rate is consistent, because it is a function of how the process is managed. The defect rate is not constant at every moment, but it drifts up and down around an average. Because of the non-constant defect rate, the i-chart is the best chart to show time dependent performance.
OK, but what if the data is truly a perfect binomial, is the i-chart still a good choice? This is the question I asked myself this morning. To answer it, I used Minitab and simulated a 10 step process where each step and a .002 defect probability for 1000 lots of 300 units. I summed up the number of defects produced at each step to represent the total defectives per lot. This data was plotted with a p-chart, c-chart, and an Individuals chart of the defect rate. This is a true binomial process, so if all three charts show the same out of control conditions, then I can believe the i-chart is also acceptable when the p-chart assumptions are met.
Here is the p-chart of the data
p-chart
Here is the c-chart of the data
C-chart
Here is the i-chart
Individuals chart
As you can see, all three charts provide the identical insight into the process behavior, three points above the upper control limit. Now these are all random/common cause events, but 4 out of 1000 data points is reasonable for 3 sigma limits. This demonstrates that the choice of an I-chart is appropriate for defect rates, even if the p-chart assumptions are met.
Join the legion of practitioners working to eliminate the p-chart from scorecards and performance reporting.
Please comment if you have also experienced p-charts reporting OOC when the process was really predictable.
The next logical step would be to use a X-bar chart. This I chart shows 4 out of control points, which according to Shewhart should be investigated. Since these data were randonly generated we know that they are false alarms. With 1000 individual observations, your expected number of observarions to fall outside the control limits would be around 3 (.9973*1000). X-bar (with the accompanying S chart) would eliminate or reduce these false alarms.
Not to be a party-pooper, but I respectfully strongly disagree and encourage readers to stick to better methods. As a long-time expert in SPC (researcher, professor, consultant, mathematical statistician), in particular I urge informed readers to avoid inappropriate uses of the so-called “individuals” chart. In almost all cases there is a better choice, often much better, as can be shown via statistics or computer simulation. My advice is to use this “I” chart sparingly, and only when you’re in a real pinch with no other easy option. It’s introduction to the SPC community is unfortunate, and it’s continued widespread adoption is more unfortunate still.
To thoroughly explain the several problems would take a longer reply that this and some tangent into mathematics, but one important insight is this: It is not so much whether these charts perform the same WHEN WE KNOW THERE IS NO PROBLEM - the question is how do their performances compare when the process initially or later may or may not be out of control, unknown to us (ie, the purpose of control charts). If we already knew the process was stable and remained so, of course there’d be no need for the chart in the first place… Quoting Deming, “there is a better way”. Ultimately the choice is yours, just my two cents. Best regards.
Thanks for the comments. Every thoughtful opinion will be posted.
As for Keith, I am not sure that an x-bar-r chart would ever be appropriate in this situation since the data was not collected in a subgrouped manner and does not have a natural time period subgrouping. Generally, subgrouping based on anything other than an underlying demographic of the data is dangerous.
Now for the comment from James. I respect your background, but my background includes the same degrees and experience but also includes many years as a practitioner of SPC in a manufacturing environment. The use of an individuals chart in place of the full range of Shewhart charts has both a theoretical and practical basis of which you do not know of or are discounting. Donald Wheeler would argue against your point with more passion than you might expect from a statisitician.
In my experience, I find the Shewhart chart selection rules fine for the observation of a process. They are excellent at identifying assignable causes, as Shewhart intended. But they begin to generate excessive false special cause indications when they are used outside of what Shewhart intended. I see the p-chart as a perfect example. When used on random data it looks like a valid chart. When used on real data is it nearly always out of control. Why, because the p-chart is out of control if the data does not match a perfect binomial distribution, since that assumption is the basis of the control limits. If just two different defect rates are included into the data and they are not probable at the same exact rate for every item or transaction, the p-chart is out of control. That is why most practitioners have learned that the p in p-chart means “Probably should use an individuals chart”.
In a complex factory running many “similar” but not matched tools, and many products, the number of SPC charts can grow to over 10K with use of computerized SPC systems, and computer-integration of gage data feeding multiple chart types from the same measurements. This way, operators can get actions from SOME rules and SOME charts that are appropriate for operator action, and engineers can get alarms showing degree of mismatching of tools and gages, for example.
Even machine sensor data is now studied for fault detection prior to adaptive control tuning based on metrology data, all automatically collected, and with graphical displays for engineers and operators. If machine has too many sensor faults, do not use the metrology results for adjustment, instead stop and fix the tool. More than half of the process tools in a high tech factory now have FDC and R2R adaptive control systems fed automatically from sensors and metrology tools with high sampling rates compared to daily PM check data in older factories.
In my experience LIMITS are almost always WRONG using X-bar-R or X-bar-S charts when sampling is done for subsequent DIAGNOSTIC purposes, not for optimal false alarm rate. Example: 9 sites on a semiconductor wafer help maintenance teams see zones on wafers that have tool conditioning issues, but most of these 9 sites are correlated when all is well, leading to wrong limits since Shewhart method using R or S for X-Bar limits requires uncorrelated sites. It is not helpful for the consultant to tell the engineer he should use only 2 sites (either for lean or sigma rationale) when his actions require diagnostic information that is almost free with automated gages.
Similar issues with lot to lot autocorrelation are not best charted daily and some suggest (just to avoid autocorrelation), as cost of material is extremely high and tools may shift, but this frequent sampling requires that engineer avoids many of the Shewhart run rules and yet catch small shifts. Box and Hunter suggested EWMA or CUSUM charts PLUS the original Individual or Xbar chart to help diagnosticians figure out when to ADJUST for tool wear or shifting. And often engineers use a CHART PER TOOL, plus an overall CHART PER PROCESS STEP across tools…from same data acquistion system.
Management metrics are another story altogether. No scrap risk on lot by lot basis, just overall bonus paid or not in many cases, and big picture fuzzy averages for the illusion of control in many cases.
Both types of charting, costly manufacturing or big picture management measurements, require up front process cost and behavior characterization and variance components analysis prior to picking a sampling plan, and some kind of adjustment algorithm or action plan linked to each of the rules used to trigger action from any chart. The FED for example, has many charts with rules and diagnostic capabilities, as do hedge fund managers. EWMA types seem to be very common there, as well as multivariate charts. Shewhart charts may be used, but not alone, in my experience. And as we said, SAME data can go to many charts with many rules that have different actions (diagnose or adjust or increase sample rate).
In general, Shewhart chart types and limits methods are not appropriate for many highly automated manufacturing operations, where rich data is available for more effective decision support at low sampling cost but high economic risk for small shifts and drifts, in my opinion.
Another example of mis-use of P charts is when 100% testing is used on high volume part manufacturing, and yield or fail bin percentages involve thousands of parts per lot, and some automated SPC systems calculate the percentages from the huge counts and then apply LIMITS based on “sample” size. Most lots are then OOC as you have all seen. in reality, I charts are used and limits set by percentile methods to trigger team actions, or to ship lots for high reliabilty vs commodity applications instead of p charts. But some systems default to p charts for this kind of data unless engineer cheats the system. And some SPC chart systems have no way to manually set limits…and vendors are proud of that “discipline.”