## 2020

#### Keeping Your Charts Under Control

Little known ways to use control charts are a valuable resource

by Philip Stein

While in the process of preparing the ASQ Statistics Division's 2002 special publication on statistical thinking in measurement, I ran into an interesting issue I'd like to tell you about here. First, some background.

Many, if not most, quality practitioners who deal with quantitative matters have seen and probably used the Shewhart control chart. This simple, extraordinarily powerful tool is usually used as a statistical filter, especially in situations in which process feedback is employed. You may not have realized your application of control charts could be described by what I just said, but it's probably true.

Control charts are used to monitor processes to see if they have changed or sometimes, as we will discuss later, to see if they are changing the way they are supposed to. A process is something that is supposed to happen the same way every time, but it will inevitably experience variation. Some of this variation, such as the choice of color applied to an object when it's painted, is planned. Other variation is unplanned and usually unwanted.

Process control is used to address unplanned variation by measuring some parameter that indicates proper process operation and taking appropriate corrective action (feedback) when that parameter changes in undesired ways. Most process variations, however, are small enough not to disturb the process outcome but will still exercise the feedback mechanism and may actually increase variation. Small or not, process variation that is consistent and random--what we usually call common cause variation--cannot be reduced by feedback. Feedback is appropriate, though, to address special cause variation--changes that have systematic, identifiable causes.

That's where the control chart comes in. It is the tool, the statistical filter, that enables us to differentiate between special and common causes. We can then feed back corrective action to adjust for special causes without the increase in random variation that would come from trying to feed back inappropriate corrections from common causes.

What makes a good control chart?

The classic control chart has several well-known and widely taught characteristics. A centerline and control limits (at three sigma) are calculated according to published formulas, and certain rules are used for reading and interpreting the patterns that appear on the chart. If the parameters of one of the rules are exceeded (violated), this is evidence the process being monitored is out of control. This indicates the data are no longer random, a special cause of variation should at least be suspected, and corrective action is called for.

The typical set of rules is adjusted for a risk of about one in a thousand (0.1%). This means truly random variation will set off each control alarm about once in a thousand data points. This would, of course, be a false alarm, since the underlying data were actually random and no special cause existed.

These rules, and the details of how to make a chart, have been so widely taught and published they seem cast in stone. Most quality practitioners without advanced statistical training would never question the formulas for control limits, nor would they allow exceptions to the sets of rules used for interpretation.

In fact, the three sigma limits and rules for control violations are simply good general purpose choices. They are not inherent truths, as some people and some publications hold. Using wider limits and the same rules, for example, will produce a chart with less sensitivity to process changes, but with a lower false alarm rate. Narrower limits will increase sensitivity and will also set off more inappropriate warnings. The limits and rules everyone has become so accustomed to are just good trade-offs between sensitivity and false alarms. Statistical literature is full of papers discussing the theory and merits of different choices of trade-offs, different means of calculating limits and entirely different ways of plotting the charts themselves.

The details of how to make and interpret control charts are also based on certain assumptions about the underlying process being monitored. We assume our desire is for the process to remain stationary, stable and constant, and the proper state of affairs is for the random unplanned variation to be distributed as a Gaussian bell curve. If these assumptions aren't true, it's still possible and productive to use control charts, but the limits, rules and even the construction of the chart itself may need to be different from the usual ones found in textbooks.

Ways to use other control charts

Metrology is one area in which different, unusual and even creative charts are particularly useful. The underlying distribution of expected variation is not always Gaussian. Sometimes, a measurement process is known to exhibit spikes, where occasional outliers do not represent textbook problems but rather should be expected and ignored. It may be particularly important to be extra sensitive to the drift of the mean while being less sensitive to changes in short-term noise. Custom control charts can easily accommodate all these requirements and more; they even appear like ordinary charts unless you look closely. When you are using common control charts in some of these different ways, it's probably best to document these differences so as not to worry readers and users who are familiar only with what's in the books.

Control charts can even be used on nonstationary processes. A manufacturing requirement might call for an oven to execute a temperature profile that changes with time: ramping up, holding and cooling down at designed rates for specified times. The actual oven temperature will not follow those instructions exactly, though. There will be random and perhaps systematic deviations from the ideal, and these can be monitored and detected by a special purpose control chart. Such a chart won't look much like anything in a statistics text, but the principles are exactly the same, and the chart is as useful as it would be in a more classic situation.

Metrologists and other advanced quality engineers have left the strictures of conventional learning behind and are applying control charts in new and creative ways. I invite you to think of doing the same.

Note: If you haven't seen the Statistics Division's 2002 special publication on statistical thinking in measurement, it can be ordered from ASQ. Ask for item number S1010.

PHILIP STEIN is a metrology and quality consultant in private practice in Pennington, NJ. He holds a master's degree in measurement science from the George Washington University in Washington, DC, and is an ASQ Fellow.

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