# Cmk and Cpk - When will we need to calculate Cmk? Machine Capability

S

#### Stevenli

Dear all,

When will we need to calculate Cmk? Sometimes we can't collect the equipment data, then we have to collect the data from the production characteristic to calculate the Cmk. Is it acceptable? If it's true, how to decide the tolerance?

And in this case what's the difference with Cpk, long -term, short-term?

Anybody can give me some hints would be highly appreciated.

Steven

#### Paul F. Jackson

##### Quite Involved in Discussions
Yes, it is usefull, but not widely used.

Cpm is like Cpk (or, more correctly, like Ppk) but you calculate the standard deviation from the TAGET, instead of from the AVERAGE (i.e. inside the square root you use (Xi-T)^2 instead of (Xi-Xbar)^2).

When the process is centered, Pp, Ppk and Cpm are the same (I am assuming that the center of the specification is the target, what is NOT allways the case)

Pp does not change when off-centering (it just compares specification against the variation arround the process average.

Ppk gets lower when off-center but does not take the target into account. It compares the distance between the average and the specification limit against the variation arround the process average.

Cpm gets lower when off-target. It compares the specification against the variation arround the target.

You can think of the variation arround the target as the varieton of the individuals arround the process average plus the distance between the process average and the target (just the concept, not mathematically correct).

For example, imagine that you have a process that you want to run at Ppk 1.5. That is that the process average is at 4.5 sigmas form the closest specification limit. If you improve the process variation (reduce sigma) you can move the average closer to the specification limit (to keep 4.5 sigmas) and the Ppk is the same. However, the Cpm would have worsened, because the average distance of the individuals from the target would have increased due to the "distance between the process averaage and the target" term.

We could say that Cpm is more "loss function" (aim to the target) focused, while Ppk is more "goal post" (meet the specification) focused.

Now, as Stan said, if "zero" is a phisycal lower limit (like in ovality, rughness, etc.), then Cpk or Ppk may be more suitable (using the upper specification limit). Even in that case, Cpm can give you extra information.

Let's take ovality as an example. Let's say that the specifiecation is 0.1 max. Just to simplify the example, let's say that the ovality is normally distributed (which most probably will NOT be the case). Let's say that you current process delivers an average ovality of 0.05 with a standard deviation of 0.02. That leads to a Ppk=0.83 and a Cpm=0.31.

Now let's say taht you make a change in the process with which you can cut variation to 0.01, but the varage ovaliti goes to 0.07. Now you got a Ppk=1 and a Cpm=0.24. Leave the absolute vaues and think of the change.

The Cpk has improve because the "worst case" parts are "more in specification". Average + 3S = 1.1 before and 1 after. You will have more parts inside the specification.

The Cpm worsens because, even with an improving contribution due the reduced variation, the worsening contribution due to the larger of-target increase has more weight. You want "zero" ovality, on average you had 0.5 before and you have 0.7 now. You will have more parts farther from the target.

So, you seem to have a god capability: "My current process deviates a maximum of less than one half of the tolerance width". Maybe you should use the Cpk or Ppk to inform your customer about your ability to meet the specification and monitor the evolution of Cpm (regardless its absolute value) to direct your improvement efforts. When you improve Cpm you ALLWAYS improve Cpk.

From the link at the beginning of this post:

"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 most accurately depicts this."

Last edited:
A

#### Atul Khandekar

Stevenli said:
Sometimes we can't collect the equipment data, then we have to collect the data from the production characteristic to calculate the Cmk.
Steven,
I haven't clearly understood what you mean by this statement. I believe Cmk pertains to machine capability and the data does come from a production characteristic for which specs or tolerance limits would be defined. you need to ensure that the variation in the process is due to machine alone.

Cmk is calculated just as Ppk. The idea is that you must do what you can to exclude every source of variation except the machine itself (same operator, same gauge, same enviroment, same raw material, nearly same time). To do so, the ussual procedure is to let the process stabilize and then collect the sample with all consecutive parts. Be careful, you shold keep the right order and avoid mixing the parts to assure that the subgroups are actually in the same order as their parts where produced. In that way you can plot a control chart, asses for stability, and calculate Cm/Cmk just as if it was Pp/Ppk.

A lot of confusion exists about capability metrics because different companies use different symbols to define various facets of capability. Before you can truly understand what a particular index means, you must look at its formula as well as how the data are supposed to be collected for its calculation.

Having given that disclaimer, the Ppk index is typically used to measure short-term process performance capability. The formula is as follows:

Ppk = Minimum (Ppl, Ppu)

where Ppl = (mu - LSL) / (3 sigmaST) and Ppu = (USL - mu) / (3 sigmaST)

Here, mu is the process average (estimated from X-double bar) and sigmaST is the short-term standard deviation, estimated from RMS formula.

I have seen the Cmk index defined in two different ways. In the first, it is used to measure machine capability (the "m" stands for "machine"). Its formula is similar to the one given above for Ppk, but the data used to estimated mu and sigmaST come from a special study conducted in such a manner as to focus on only process variation originating from the machine under study. Thus, the same operator would be used for the entire study, the same batch of material, the same setup, etc. Be aware that some companies will replace the 3 sigmaST with 4 sigmaST.

In the second definition I have seen, Cmk is used to quantify the performance of a measurement system, so here the "m" stands for "measurement." Again, its definition is similar to that given above for Ppk, but the data come from a properly conducted gage study. To further complicate the issue, some companies use 5 sigmaST in place of the 3 sigmaST that is normally used.

As you can see, to really appreciate the difference between any two capability metrics, you must first obtain the formulas and then an understanding of how the data are to be collected. Without knowing this, there really is no way to correcttly answer the question "What's the difference between Cpk and Cmk?".

By the way, I have also seen the following indexes: Cm, Cpm, and Cpmk !

Hope this helps.

S

#### Stevenli

Hi, Atul

Because some consultants said if we can collect the data from the machine itself, then it's better to calculate the Cmk. Otherwise we can collect production characterisitic data to get Cmk.

According your post and reference if I ensure the process variation only from machine, then can I say Cmk is equal to short-term Cpk?

Best regards,
Steven

G

#### gandhi.v

I am new to this.
But can any one tell that how it differs if we use 3sigmaST/4SigmaST/5Sigma ST.