Statistical Process Control, The Alpha and Omega of Six Sigma, Part 3: Process Capability and Six Sigma

Rip Stauffer

Process Capability

This is the third in a four-part series on Statistical Process Control. In the first segment, we looked into the history behind Statistical Process Control; in the second, we examined a few of the basic tools. In this segment, we will look at concepts associated with an aspect of process performance whose measurement Statistical Process Control makes possible: capability. Process capability is the theoretical foundation for Six Sigma metrics, so we will also explore that relationship.

In a 1934 paper, Walter Shewhart (the father of Statistical Process Control) defined three types of quality1:

1. That quality that characterizes a thing itself independent of all other things and of human volition and interest.
2. That that characterizes a thing A in its relation to another thing B as a part of a whole and independent of human volition and interest.
3. That quality that makes a thing wantable by some one or more persons.

In the same paper, Shewhart pointed out the limitations of these types of quality. Type one is fairly easy to deal with. Some specifications may be set, based on desired quality characteristics, and machines or producing processes controlled to keep the output consistent with those specifications. Type two quality characteristics may also be measured as outputs of systems. These measurements can be tracked using process behavior charts to characterize the Voice of the Process (VOP).

Type three is trickier; it requires that we tap into the Voice of the Customer (VOC), articulate that voice as a set of measurable characteristics, and then translate those Critical-to-Quality (CTQ) characteristics into process measures. Essentially, this translation should match the two voices: VOC and VOP. The matching of these two voices is done via the concept of process capability. This process capability concept is fundamental and definitive in Six Sigma; the term "Six Sigma" came from capability concepts and studies, and process metrics such as DPMO and the Process Sigma come directly from process capability concepts.

The basic idea of process capability is actually simple; it’s all about "elbow room." If we have a stable process, then virtually all of the process output will be contained within three sigma of the process mean. If we convert the customer specifications to sigma units, we can compare the width of the process to the width of the specification limits.

Figure 1 illustrates this.2 We have a process that is, essentially, Six Sigma units wide; we have specification limits that are 12 measurement units wide. In this case, each sigma is exactly two measurement units wide, so the specification limits are also Six Sigma units wide. Note the process capability indices, Cp and Cpk. In this case, since the process is centered in on the target, the specification limits are Six Sigma units wide and the distance from the process mean to the nearest specification limit is three sigma units wide both Cp and Cpk are equal to one.

Figure 1—Capability

This is not a bad place to be. It beats many of the alternatives. You could, for instance, have an unstable process—you can’t manage it because you can’t say anything about what it might do. It also beats the incapable states depicted in Figures 2 and 3.3 In Figure 2, the specifications are only four sigma wide, so you can predict some percentage of output either smaller or larger than specified. In Figure 3, the specifications are still Six Sigma units wide, but the mean has shifted, leaving 2 sigma hanging outside the lower spec. So it’s not bad—in the Figure 1 case, we can predict that we will continue to meet customer specifications almost 100 percent of the time, as long as the process remains stable and we don’t let the mean shift.

Lack of Capability (Variation too Wide)
Figure 2—Lack of Capability (Variation too Wide)

Lack of Capability (Off -Center) 
Figure 3—Lack of Capability (Off -Center)

The Figure 1 case does have its limitations, though. It is a little like driving down the highway between concrete construction barriers that are placed an inch or two wider than your vehicle…you can continue to drive without damaging anything, but you have to be very careful. It’s much better to have some more elbow room, some safety room. In Figure 4,4 we have what you might call idealized Six Sigma quality…the process mean is centered on the target value, the sigma units are one measurement unit wide, and so there are six sigma units from the process mean to the nearest specification limit. In this case, we have a predictable process that uses half the customer-specified tolerance. As long as the process remains stable, we will be very unlikely to see any non-conforming product or service. (Two occurrences out of a billion; 3.4 out of a million if you factor in the Motorola 1.5-sigma undetected shift.)

Six Sigma Quality
Figure 4—"Six Sigma" Quality

This case also gives us what has come to be called the "area of robustness" illustrated in Figure 5.5 In Figure 5, the variation has not changed, but the process mean has shifted to two sigma units above target. Since our specification limits are still 12 sigma wide, though, the nearest specification to the mean is still four sigma units away. Therefore, we still have a very small probability that any out-of-spec output will happen. Increasing elbow room obviously has its advantages. (Click on diagram to enlarge.)

Area of Robustness
Figure 5—"Area of Robustness"

Process Capability and the Taguchi Loss Function

In the first segment, we introduced the Taguchi Loss Function. The idea that there is some loss as we move away from the target value toward one of the specifications is no mystery to anyone who has worked with tolerance stack. Taguchi’s concept has some far-reaching implications for quality, though; consider the two cases6 in Figure 6.

In the left hand side, distribution A is from the Figure 1 case, distribution B is from the Figure 4 case. With B, almost 100 percent of the output is produces with virtually no loss, with around 70 percent produced at almost zero. If you have A, around 70 percent of your output will be produced with the same loss as all of B; with some in the tails approaching very significant loss. On the right-hand side, the variation is equal, but the mean has shifted. In this case, most of the output is coming from an area where the curve is growing; very little is produced at or near zero-loss. In this case, operating in the "area of robustness" generates considerably more loss than operating on-target with more variation, as in distribution A.

So the Taguchi Loss Function concept suggests that yes, we need to always reduce variation; but we also need to get the process on target and keep it there. It answers the question, "What do I work on first?" Of course, first you have to get the process stable; once that’s done, get it on target and keep it there while you continually work to reduce the variation. (Click on diagram to enlarge.)

Impact of Taguchi Loss Function
Figure 6—Impact of Taguchi Loss Function

Summary of the Exploration of Process Capability Concepts

We have looked at the customer-focused side of Statistical Process Control, exploring process capability concepts. Process capability concepts arise from Statistical Process Control; a control chart of individuals shows us the raw capability of a process, and a capability study requires a stable process. We also looked at the Taguchi Loss Function, the concept that drives us to strive for Six Sigma levels of performance.

In the next segment, we’ll discuss some of the methods needed to put these process capability concepts into practice in Six Sigma. We’ll talk about how process capability concepts contribute to Six Sigma metrics, and discuss the use of Statistical Process Control in the DMAIC project life-cycle.

1. Shewhart, W. A. (1934). Some aspects of quality control. Mechanical Engineering, 56(12), 725-730.
2. Stauffer, R. F. (2008). Woodside Quality Solutions Black Belt Course Manual. Minneapolis: WQS.
3. DeZouche, C., Mehaffey, R. A., & Stauffer, R. F. (1998). Systems Approach to Process Improvement (2nd Ed.). Washington, D.C.: NPRDC.
4. Ibid.
5. Ibid.
6. Ibid.

Continue to part 4 of this series.