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Scott A. Hindle


Process Capability: What It Is and How It Helps, Part 3

There’s more to it than Cp and Cpk numbers

Published: Wednesday, August 31, 2016 - 15:40

Part two of this four-part series on process capability concluded with Alan just about to meet Sarah for a second time. He thought he was making good progress with his analysis of Product 874 data until he was asked to assess process capability, even though it can’t be assessed for an unstable process.

Making sense of the XmR chart

Alan thanked Sarah for the two articles she’d given him. He said that, guided by the second article by Donald Wheeler, he’d created his first XmR chart (figure 1 below), which he interpreted to mean that the process data represented an unstable or not-in-control process. Wheeler’s article noted that it was more important to find the cause of process changes rather than computing statistics.

Figure 1: Alan's XmR chart for Product 874 data. Click here for larger image.

Sarah was impressed with Alan’s progress. Her first question was if the data he’d used to create the XmR chart were in their time-order of production sequence. Alan wasn’t sure. He asked why this was important.

“Probably the most important piece of contextual information in any set of process data is its time-order sequence,” Sarah responded. “This can have a major influence on both the ‘picture’ the data create on an XmR chart and the standard deviation coming from the moving ranges. The time-order sequence is lost in histograms and box-plots. It can also be lost in many statistics, like the global standard deviation statistic—i.e., ‘STDEV’ in Excel, applied to all the data. Once you’ve lost the time-order in the data, you’ve lost a lot of the useful information. Assignable causes, or process changes, often escape detection once the time order is lost.

Alan opened the original email in which Product 874 data were found. One of the worksheets in the Excel file of the data contained a column labeled Date/Time. This confirmed that the data were in the time-series of production. Another column, Production Order, caught Sarah’s attention. This confirmed that the data values were a mixture of within- and between-production runs. (Sarah interpreted a production order to be a “production run.” Different production runs were on different days.)

Alan asked why this was important. Sarah commented that such factors can play a key role in the way data are organized and in determining the most appropriate chart to assess process stability. After 10 minutes, they had created the data chart in figure 2 as a better way of representing the capability data.

Figure 2: Sample data for a product characteristic in Product 874. Click here for larger image.

Sarah was able to use figure 2 to help Alan understand that:
• Most of the moving ranges account for within-production-run variation (e.g., the first three moving ranges of 1.22, 0.86, and 0.41 are from within production run 1).
• With 17 total production runs, 16 of the moving ranges account for between-production-run variation (e.g., the fourth moving range of 0.37 is the difference between the last value of production run 1 and the first value of production run 2).

Hence, Alan's XmR chart made it possible to looks at the stability of within- and between-production-run variation. Sarah commented that an average and range chart could be considered for these data, but one complication could be a nonconstant subgroup size, which leads to varying limits on this chart. They agreed to work with the simpler XmR chart, at least for now.

Sarah proposed to update Alan’s XmR chart by labeling the x-axis “production run” rather than “observation number.” They both agreed that the result, shown in figure 3, improved the picture of the data.

Figure 3: XmR chart for Product 874 data with the x-axis based on production run. Click here for larger image.

Alan knew from his own XmR chart that the points signaled with a “1” were those outside the process limits. He didn't know what the two values with a “2” next to them signified. Sarah asked if Alan would expect nine consecutive values to fall on one side of the average. He said no but didn’t sound convincing. Sarah then said that this is like flicking a coin nine times and always getting a head. It wouldn’t be a natural or expected outcome, given a fair coin. This detection rule, she explained, is used to detect small but sustained shifts, with Rule “1” detecting the big signals of change.

Sarah then asked if Alan could see any signals of process change in figure 3, or what she likes to call “surprises,” aside from those picked up by the standard detection rules. Alan couldn’t see any, and Sarah agreed.

The link between the chart and process capability

Alan asked Sarah what all this has to do with process capability. He’d already calculated Cp and Cpk to be 1.19 (see part 1). He stressed again that he had been left confused by the statement in Carl Berardinelli’s article, “A process should be stable and in control before process capability is assessed.”

Sarah opened a book she had at hand and asked Alan to read the following text:

“Process capability is a prediction of future performance of the process. It can be stated as the range of outcomes expected for any measure of interest. Before considering capability, the process for the measure of interest must be stable (no special causes). This gives us a rational basis for the prediction. Thus, developing an appropriate Shewhart chart for the measure of interest is a prelude to a capability analysis. The process capability can be compared to the requirements or specifications for the measure. This is best done graphically.”

Sarah said that an assignable cause is a special cause. She then drew two arrows on her X chart, which now looked like figure 4, and asked Alan what degree of belief he’d have in future process output falling inside the red arrows (the limits).

Figure 4: X chart for Product 874 data with the x-axis based on production run. Click here for larger image.

Alan said that most of the data fall comfortably inside the limits. He’d have a reasonable degree of belief.

Sarah commented that, for a stable or in-control process, approaching 100 percent of process output is expected to fall inside the limits. “With two of 56 values outside the limits (which is 3.6 percent), and given the run of nine points above the central line, the data behave inconsistently with this expectation,” she said. “The detected process changes argue against confidently predicting within limits—which are also called control limits or, given an XmR chart, natural process limits.”

Sarah asked Alan how he could claim that these causes of process change, or instability, wouldn't return. “The chart detects assignable-cause variation, and until that is identified and effectively controlled, the causes of process change may well strike again, and the magnitude of their effect would be unpredictable,” she said.

For this reason she characterized the process behind Product 874 data as unpredictable. She preferred to speak of “predictable” and “unpredictable” processes because predicting what an “unpredictable process” will do is something people can readily understand as a contradiction.

Sarah then turned a printout of figure 4, 90° to the left to help Alan visualize what Cp is all about. Her completed picture with annotations is shown in figure 5. “It may be helpful to think of the process as a car (i.e., the red lines) that you want to park in a garage (the blue lines),” she said. “Moreover, this picture is usually created with a histogram of the individual measurements.”

Figure 5: Rotated X chart to visualize the way Cp is quantified and interpreted. Click here for larger image.

Sarah stressed that approach two, using a ruler, is offered to better understand what the mathematical formula does. The most important question centers on the interpretation of how trustworthy a capability statistic is, which introduces the dependency on the XmR chart when using individual values:
• Predictable processes: Cp and Cpk can be considered reliable indicators of future performance. (Note: If planned changes to a process will take place, these may undermine the reliability of Cp and Cpk as indicators of future performance.)
• Unpredictable processes: Cp and Cpk may be false, or very misleading, indicators of what the process will give in the future (i.e., what you expect may be quite different to what you get from the process in practice).

Alan said he thought he understood this, but it would be great if Sarah could give him a simple analogy that he wouldn’t forget and could also use when working with others.

“Think of a capability statistic to be like a secondhand car you’re thinking of buying,” said Sarah. “Someone who you don’t really know has told you of just the car you’re looking for, and the price is one you’re willing to pay. Would you buy it if this is all you knew, or would you ask a mechanic to look over it first? Assessing the predictability of a process before doing capability is comparable to having a mechanic check over the car before you hand over the money. The mechanic telling you everything is in order is like a characterization of process behavior as predictable. You know what you’re dealing with; you’re looking at capability statistics you can place some faith in.”

Alan thanked Sarah for the analogy but said he now felt a bit deflated. He knew that his original capability statistics didn’t quite mean what he thought they meant. As Wheeler suggested, now was the time to question what had caused the detected process changes.

Different standard deviations, performance, and capability

Looking again at Sarah’s illustration of Cp (figure 5), Alan suddenly realized her Cp value of 1.31 was different than his value of 1.19 (see part 1). It also dawned on him that he’d used the global standard deviation statistic in his computation, what Wheeler had labeled as “wrong” for an X chart.

Sarah said that he’d actually computed performance indexes, so he should have labeled them as Pp and Ppk, adding that these are essentially one-number descriptions of the past. Process predictability does not come into play because these computations focus on “what was” and not “what can be expected.”

In the case of individual values, the estimate of standard deviation for capability indexes comes from “average moving range” ÷ 1.128. Sarah finished by saying that capability indexes are predictive indicators, and the confidence in the prediction is dependent on the interpretation of process predictability.

Next steps

Rather than getting bogged down in more technicalities, they agreed to work on an action plan. They would:
1. Find a contact at the factory who could provide more context about the data and the production process
2. See if the assignable causes indicated by the XmR chart could be identified
3. Document their findings in a report

Alan agreed to take the lead, starting first with identifying a factory contact. More about that in part four, which will be published in tomorrow's issue.


About The Author

Scott A. Hindle’s picture

Scott A. Hindle

Scott Hindle supports R&D and factory operations on process capability studies for new products and processes, statistical process control (SPC) for use in routine production, and the use of online measurement devices as a part of both SPC and engineering process control.


add the link

hi could you add the link to part 4?

Link added

Good catch. Thanks Paul