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Note: A discussion on "What is Cpk?" is in the Video Podcast below:

Cpk vs. Ppk

by Keith M. Bower

To assess the productive capability of a process, Six Sigma practitioners may be required to perform a capability analysis. When obtaining the results from a statistical software package, process capability estimates (for example, and ) and process performance estimates (for example, and ) are typically reported. This article provides some recommendations for obtaining valid results when considering "long-term" versus "short-term" variation in Six Sigma projects.

Process Capability Indices

It is important to note that the only difference between the calculations for versus and versus is in the denominators, as shown below: 1

  1. =
  2. =
  3. =
  4. =

Where:

  • USL = Upper Specification Limit
  • LSL = Lower Specification Limit
  • μ = process mean
  • = process standard deviation based on variation within the subgroups alone
  • = process standard deviation based on overall variation

Let = the estimate of within-subgroup variation (e.g., based on ) which represents the inherent variation of the process (due to common causes), and let = the estimate of overall variation
(e.g., based on ≈ S for "large" studies).

The value can be interpreted from equation (5) as the total variation exhibited by the process in a study:

To understand the practical difference between σWithin and σOverall in capability analyses, consider the example that follows.

An Example

A critical to quality (CTQ) characteristic has exhibited statistical control around a mean of 201mm and standard deviation of 4mm. The specifications for this process are 20013mm. Rational subgroups of size 5 are taken at regular intervals to monitor the process.2 Twenty subgroups are available for analysis and an estimate of capability is required.

As shown in Figure 1, the process is in statistical control and the assumption of Normality appears reasonable.

Figure 1

Since the analysis occurs in "Phase II" of a control charting study, a mean of 201mm and within-subgroup standard deviation of 4mm are used in the chart.3 Note from Figure 2 that when computed from the data is 3.9576, which is very similar to (3.9596), and the value (4.0) entered by the user.

Figure 2

This illustrates a key fact:

When a process is in statistical control there will be little difference between and

Therefore, it would make no practical difference which standard deviation is used:

≈ and ≈

For this example note that:

  1. The approximate 95% confidence interval for is between 0.21 and 0.29
    (= 0.25).
  2. The approximate 95% confidence interval for is between 0.22 and 0.29
    (= 0.25).
  3. The approximate 95% confidence interval for is between 0.10 and 0.24
    (=0.17).
  4. The approximate 95% confidence interval for is between 0.10 and 0.23
    (=0.17). 4

Phase I/II Studies

Where I perceive some controversy to exist in the Six Sigma community is not in discussions of Phase II studies, the final stage of the DMAIC cycle, but in Phase I.

When a process is not in statistical control, may be much larger than owing to a high level of between-subgroup variation relative to the "inherent" variation. Practitioners may then be tempted to report a capability estimate using as a "worst case" scenario for the future.

Reporting this as the expected capability of the process in the future is, however, disingenuous, since the process has not exhibited stability. As W. Edwards Deming wrote: "The performance of a process that is in statistical control is predictable. A process has no measurable capability unless it is in statistical control."5

Moreover, any confidence interval statements from a process that is out of statistical control would be meaningless, since we would not be sampling from a stable system.

Short Term versus Long Term

A raw estimate of the proportion nonconforming, using historical data at the start of a Six Sigma analysis, typically has no "long-term" inferential value as a capability estimate, since the process is usually out of statistical control. It may have use, however, as a point estimate, for example, to illustrate the poor capability of a given process.

Moreover, owing to entropy, one can guarantee that such a process would not unilaterally improvequite the opposite. Sources of variation can and will affect the performance of a process over time. As discussed by George Box: "the long-term instability of processes, after standard techniques of quality control have been applied has been verified by extensive studies of process data."6

Adding the "Long-term" Shift and Drift Component

Many times, practitioners are encouraged to compute the parts per million nonconforming when in Phase I and use tables or freeware from websites that add the "1.5 Sigma shift" to a capability estimate. This is intended to express the nonconformance rate if the process were stable (that is, only common cause variation is present). The arguments in favor and against the value of 1.5 are not for this discussion.

In my opinion the use of the number 1.5 is moot; an understanding of the concept itself is far more important. Furthermore, adding a "fudge factor" to a capability estimate from an unstable system hardly seems appropriate statistical practice.

From a pragmatic perspective, it seems adequate for a Six Sigma practitioner to report that process performance may be "poor" when it is so, and that the system requires improvement. Frequently, one finds these instances to be rather obvious, and no use of "murky" statistical practices would be required, only common sense.

Summary

This article has illustrated when and why certain widely used capability indices may provide similar results. Understanding the concept of "long-term" versus "short-term" variation is crucial, and the implications can be seen in the possible misinterpretation of results from capability studies.

References

  1. Automotive Industry Action Group, Statistical Process Control (SPC), (1992): 80. Note that "hats" are added to capital C and P to represent estimates for consistency with established statistical practice, a distinction not made in the AIAG manual.
  2. For more information on rational subgroups and control charting, see Douglas C. Montgomery, An Introduction to Statistical Quality Control, 4th ed. (New Jersey: John Wiley & Sons, Inc., 2001): 170-172.
  3. For more information on Phase I and Phase II studies, see William H. Woodall, "Controversies and Contradictions in Statistical Process Control," Journal of Quality Technology 32, no. 4 (2000): 341-350.
  4. For information on confidence intervals for capability indices, see Keith M. Bower, "Confidence Intervals for Capability Indices," International Society of Six Sigma Professionals: EXTRAOrdinary Sense 2, no. 4 (2001): 6-7.
  5. W. Edwards Deming, foreword to Statistical Method from the Viewpoint of Quality Control, by Walter A. Shewhart (New York: Dover Publications, Inc., 1986), ii.
  6. George E. P. Box, "Six Sigma, Process Drift, Capability Indices, and Feedback Adjustment" Part D.1. in Box on Quality and Discovery: With Design, Control, and Robustness (New York: John Wiley and Sons, Inc., 2000), 295. N.B. Also available via Download 176 after clicking here.

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