Last night we discussed the **Taguchi Loss Function** and how Taguchi methods are more concerned with hitting the target compared to more traditional methods that often focus on keeping our data between the upper and lower specification limits.

**Cpm**

Staying with this theme I now want to introduce Taguchi’s version of **Cp and Cpk** which we call Cpm. Truth be told I don’t know where the “m” comes from. Perhaps it has ties to a Japanese word. I will default to someone like Jon Miller who knows a bit more Japanese than I. Also, if someone else knows where the “m” comes from please do share!

What I do know is why we actually prefer Cpm to Cpk in certain situations. It all has to do with the relationship between our “target” and the specification limits.

**Example of when to use Cpm**

For example, let’s say we are machining a part with a LSL (lower spec limit) of 10 mm and a USL (upper spec limit) of 20 mm. Let’s also assume our customer asks us to produce the part to 18 mm instead of the more typical 15 mm (dead center).

In other words, instead of being right between the LSL and USL our customer wants us to bias the dimension more towards the USL. While this biasing may not be the norm, it does occur. When I teach Cpm and ask a room of 25 for examples of where they have seen a bias towards one spec limit I normally get several examples. If you have an example please leave us a comment.

**Cpk Makes no Sense**

In this situation using Cpk makes little sense since we are purposely shifting the mean of our process a bit towards the upper specification limit per our customers request. In fact, assuming we are successful and are able to consistently produce a machined part of 18 mm Cpk would penalize us since the process is “shifted.”

A better method, in this example, would be to use Cpm which uses the “target value” in its calculation. In other words we will not be penalized for not being dead center between the upper and lower specification limit.

**Math Geek Fix**

For those interested in the math, the key difference between Cpm and Cpk has to do with the the way standard deviation is calculated. The traditional Cpk standard deviation is calculated by comparing each data point to the mean of the process. When calculating Cpm we use a different method to calculate the standard deviation. Instead of comparing the data points to the mean of the process we compare it to the target value.

Note that if the target for our process is dead centered between the LSL and USL Cpm and Cpk will be almost identical.

Well that is Cpm in a nutshell. I can’t promise a third straight day of Taguchi fun… then again you never can tell. So please stay tuned!

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## 7 Comments

## robert

April 17, 2007 - 12:17 amRon

Great post summarising an often underused or at least not well understood capability metric. This is a nice link explaining the metric further, with some good examples: http://tinyurl.com/2j3ylu

Rob

## Ron Pereira

April 17, 2007 - 7:02 amThanks Rob. Cheers!

## Kim Niles

March 27, 2008 - 9:05 amRon: I’ve read that the m stands for machine … you asked. Apparently some logic suggests that if we keep individual machines set more capable than the rest of the process then the overall process capability will remain high. KN

## Ron Pereira

March 27, 2008 - 7:42 pmVery good to know, Kim. Thanks for sharing!

## Gagandeep S. Datta

August 19, 2009 - 5:12 amRon,

I have the answer for you on, “Truth be told I don’t know where the “m” comes from” … In the process capability index statistic Cpm, “m” is the total number of study subgroups which go in the calculation of the Cpm statistic.

The traditional variable control charts are to monitor the “centralizing” and “dispersing” tendency of specific quality variable in the process through sampling. When we sample basis rational subgroups, the statistic of each subgroup is the “mean” and “range” or the “mean” and “variance” of observations from a subgroup.

In traditional variable control charts occasionally, however, because of production problems, or inspection mistakes will cause a difference in the sample numbers within each sub-group. If such situation occurs, the control limits of a traditional control chart will vary with the sub-group size (i.e., as the subgroup size increases, the control limits become narrower; as the subgroup size decreases, the control limits become wider apart). Nevertheless, there will be no such problem faced if we use the ‘PCI’ based control chart, which is based on Cpm statistic.

Cheers,

Gagandeep

## Raj

October 23, 2010 - 11:14 pmA very good article.

Cheers!

## Richard West

April 5, 2013 - 2:28 pmThe “M” in CPM means 1000. This is roman letter used to indicate 1,000.

It can be confusing because in some industries M is used for 1,000 while in others M is used for 1 million (and they use K for 1,000).