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# The capability index dilemma: Cpk, Ppk, or Cpm

Lori, one of our customers, phoned to ask if Cpk is the best statistic to use in a process that slits metal to exacting widths. As a technical support analyst, I too wondered what index would be best suited for her application. Perhaps Cpk, Ppk, Cpm, or some other index offers the best means of reporting the capability of her product or process. Each of these capability indices can be calculated using software such as SQCpack.

Lori’s process capability index, Cpk, has never dipped below 2 and typically averages above 3. Given this high degree of capability, she might consider reducing variation about the target. While the Cpk and Ppk are well accepted and commonly-used indices, they may not provide as much information as Lori needs to continue to improve the process. This is especially true if the target is not the mid-point of specifications.

Cpm incorporates the target when calculating the standard deviation. Like the sigma of the individuals formula, compares each observation to a reference value. However, instead of comparing the data to the mean, the data is compared to the target. These differences are squared. Thus any observation that is different from the target observation will increase the standard deviation.

As this difference increases, so does the Cpm. And as this index becomes larger, the Cpm gets smaller. If the difference between the data and the target is small, so too is the sigma. And as this sigma gets smaller, the Cpm index becomes larger. The higher the Cpm index, the better the process, as shown in the diagrams below.

 In these 3 charts the process is the same, but as the process becomes more centered, the Cpm gets better. This Cpm is good. This Cpm is better. This Cpm is best.

 In these 3 charts, the process stays centered about the target, but as the variation is reduced, the Cpm gets better. This Cpm is good. This Cpm is better. This Cpm is best.

We can use Lori’s raw data to provide an example of how Cpm is calculated:

 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9 obs 1 90.741 102.711 104.066 106.602 100.904 104.922 112.738 102.388 97.825 obs 2 102.300 100.882 105.620 95.978 108.558 100.243 108.145 104.159 95.209 obs 3 98.642 103.314 96.165 96.265 94.882 97.053 98.679 100.204 91.273 obs 4 106.069 98.569 100.412 95.869 98.573 111.042 103.788 99.328 93.430 obs 5 97.635 96.639 96.316 84.872 108.588 99.068 105.664 94.157 98.263

And the specifications are: USL = 145, Target = 105, LSL = 60

Cpm = (145 – 60) / (6 * )

Cpm = 1.91 (Cpk = 2.51)

In a process with both upper and lower specifications, the target is typically the midpoint of these. When such a high degree of capability exists, one may want to ask the customer if the target value is ideal. Lori should check with her customer to determine if he or she wants a small shift toward one of the specifications. Regardless of the target in relation to the specifications, the focus should always be on making the product to target with minimum variation. Cpm is the capability index that accurately depicts this.

Reference: L.J. Chan, S.K. Cheng, and F.A. Spiring, “A New Measure of Process Capability: Cpm,” Journal of Quality Technology, Vol.. 20, No. 3, July, 1989, p. 16.

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