The Ubiquitous Cpk
by Lynne B. Hare
A colleague and I were discussing his long-term role at his company. He explained he was in charge of maintaining the computer banks of key process indicators (KPIs), Cp, Cpk and so on. While I am certain his function was more extensive than that, it did seem odd that he would put quality indexes at the top of his list.
The more I thought about it, the more jealous I became of the popularity and ubiquitous nature of Cpk. After all, I’ve spent a career encouraging people to “always, always, always, without exception, plot the data—and look at the plot,” to “be sure to take the right amount of the right kind of data” and to understand “the gold is in the interactions.”
But none of those admonitions has enjoyed the tenacious, pervasive popularity of Cpk. It is in the Six Sigma literature, often without question and without statements of necessary assumptions. It is included in most quality related software, quite often without cautionary words regarding its use.
For the record, Cpk is the distance from the process mean to the nearer specification limit divided by three times the capability or within-group standard deviation.
I’ve seen Cpk used for comparisons of one manufacturing line to another, and I’ve seen these indexes rolled up to form broad summary indexes to compare manufacturing facilities: the plant with the highest average Cpk received an award. In fact, when someone learned the distribution of Cpk was skewed, he “improved” plant comparisons by switching to the median Cpk in place of the mean.
But when I ask people if they understand what Cpk means, I get the deer-in-the-headlights stare accompanied by some vague words about the importance of it being greater than 1.33 for good quality. That helps me understand its appeal. Aha, Cpk is a simple, single number that has market appeal and separates good from bad. It avoids statistical gobbledygook, and its very mention dissuades resistance.
OK, it has its place then. All technologies use jargon as shortcuts, and we all play politics to some extent. Why should we differ here? Because quality is better measured by other KPIs, such as percentage within specification, first run yield, process capability and process performance, the last two speaking volumes toward opportunities for improvement.
Cpk might indicate the state of the process, but if you are really interested in process improvement, you want to know both its current state and what it could do if everything went right; that is, its capability. The difference between performance and capability is opportunity, and that drives improvement.1 Cpk often masks that opportunity, as shown in Figure 1.
So, should we outlaw the use of Cpk and its relatives, Cp, Pp, Ppk and my favorite, C∋∋∋∋? We couldn’t if our lives depended on it. They are ensconced.
We can, however, point out when its use is relatively safe and when it might lead us astray.
- First, it is important to have customer driven or functional specifications. If the specifications are not based on customer wants and needs, they tend to drift to meet political expediency: We can make Cpk look better or worse by moving the specification limits out or in. Having functional specifications makes their establishment more objective—lack of adherence to specifications might cause a nut and a bolt not to fit or the pudding not to pudd.
- The use of Cpk assumes the process is in control (or stable). This is because the denominator in its calculation is three times the capability, or within subgroup standard deviation. Ppk is an attempt to overcome this deficiency by employing a standard deviation estimate that is close to actual performance, but it has its own problems.2
- Cpk use also assumes the data follow a normal distribution. So do its competitors, you might argue, and I would agree. For some measures, I would add, normality is more an issue than for others. Some kind of verification—in the form of a normal probability plot or statistical test of normality—is in order before we plunge ahead.
- It is best to have at least 100 observations in the data set on which Cpk is calculated. This number is subjective, I know, but the amount of uncertainty associated with estimates of Cpk based on smaller sample sizes is unsettling. For example, a 95% confidence interval about a Cpk of 1.33 based on a sample size of 60 is (1.11 – 1.61), and when the sample size is increased to 120 it is (1.17 – 1.52).3 Such intervals, large and nonsymmetric as they are, make you lose faith in the index’s ability to truly separate the good from the bad.
In my experience, it is rare when all four of the above criteria are met. The reasons for failing to meet them have been exposited by others. Among the most lucid papers are those by Bert Gunter.4,5,6,7,8 The consequences associated with failing to meet the criteria can be disastrous.
If you are sincere about quality improvement or even if you simply are going to be a keeper of computer banks of KPIs, you might want to be sure you are shepherding statistics that edify.
- L.B. Hare, “Chicken Soup for Processes,” Quality Progress, Vol. 34, No. 8, 2001, pp. 76-79.
- G.C. Runger, “Robustness of Variance Estimates for Batch and Continuous Processes,” Quality Engineering, 1994, pp. 31-43.
- ASQ Chemical and Process Industries Division, Quality Assurance for the Chemical and Process Industries, second edition, Quality Press, 1999.
- B.H. Gunter, “The Use and Abuse of Cpk,” Quality Progress, Vol. 22, No. 1, 1989, pp. 72-73.
- B.H. Gunter, “The Use and Abuse of Cpk, Part 2,” Quality Progress, Vol. 22, No. 3, 1989, pp. 108-109.
- B.H. Gunter, “The Use and Abuse of Cpk, Part 3,” Quality Progress, Vol. 22, No. 5, 1989, pp. 79-80.
- B.H. Gunter, “The Use and Abuse of Cpk, Part 4,” Quality Progress, Vol. 22, No. 7, 1989, pp. 86-87.
- B.H. Gunter, “The Use and Abuse of Cpk Revisited,” Quality Progress, Vol. 24, No. 1, 1991, pp. 90-94.
LYNNE B. HARE is director of applied statistics at Kraft Foods Research in East Hanover, NJ. He received a doctorate in statistics from Rutgers University, New Brunswick, NJ. Hare is a past chairman of ASQ’s Statistics Division and a fellow of both ASQ and the American Statistical Assn.