Ppk - Capability calculation using the estimated sigma calculated from the Rbar/d2

D
#1
I have a customer who takes a 30 piece sample and determines a Ppk.
He tells me we do it wrong. This is how we do it:
We take subgroups of five, and get X bars. Then we take the X bars and average them together to get a double X bar.
Then we take the double X bar and substitute it in the Cpk formula where the X bar is. Are we doing this right? HELP!!!
 
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B

Batman

#2
Hi Dawn.
Very simply, Ppk is the capability calculation using the estimated sigma calculated from the Rbar/d2 (read Rbar devided by d2,) as opposed to the Cpk calculation using the population sigma. The d2 factor is in any look-up table, and is based on the subgroup size.

Remember, when utilizing a SPC type chart, where you determine variation over time, and thus calculate Xbar and Rbar, the Ppk calculation holds up if the plots are "in control," so 5 subgroups may be a stretch for determining an "in control" condition.

If you are taking a 30 piece sample from a lot of parts, you may consider a Cpk calculation, if the sample is random. You may even consider NOT taking that sample if the parts come from a process that has demonstrated good control and good capability.

There is much more to this, I know a couple of wizards nearby that would love to enhance this. Also I believe there is a very good Cpk / Ppk explanation in the PDF zone elsewhere in this site.
 
D

Don Winton

#3
Dawn,

Batman said it all, and there is not much I can add. Your method will satisfy the 'statistically valid' parameter, if asked, based on the information you supplied.

Regards,
Don
 
D
#4
Thanks, I will look for the definition in the pdf files.
Next, I am told if I don't use the range in a capability study, I am doing Ppk not Cpk.
Also, if I calculate sample standard deviation, I am doing Ppk . Standard deviation is Cpk. Any takers?
 
M

Mark W

#5
I don't claim to be a capability index guru, but the SPC Manual in the QS9000 documents gives a pretty clear definition of Cpk and Ppk. If using a control chart method for variable data (i.e. Xbar/R), Cpk (Capability Index) is derived from the chart with an estimate of the standard deviation taken from the control chart, s=Rbar/d2. It is used to determine long term process capability assuming a stable process and demonstrated with the control chart. The Ppk (Performance Index) is a measure of how the process is performing to the specifications based upon a sample from the population and the estimated sigma is derived using the sample standard deviation formula SqRoot(Sum(Xn-XBar)^2/(n-1)) (??or something like that??).

I believe your customer may be referring to these definitions when they state their way of calculating the Ppk index. Someone please let me know if I am way off base here.
 
B

Batman

#6
Sorry, I didn't mean to confuse, but Mark is right. 'Process Capability,' called Cpk, IS from X-r chart, and is a capability index for 'common cause' only variation, that is, from a 'stable' process. This is the Rbar/d2 calculation. Yes, Dawn, Cpk uses the range chart.

Ppk is calculated from the sample standard deviation, or also called the population standard deviation.

Ref page 80-81 QS9K SPC manual.

Mark - use the calculator, it's easier. Heh heh.
 

Kevin Mader

One of THE Original Covers!
Staff member
Admin
#7
Question:

Is the sample standard deviation (s, n-1) the same as the population standard deviation (sigma, n)? While the resultant is generally very close for averages above 25-30 samples, below you run more significant differences. When calculating Cpk, the denominator in the formula uses the 'population' estimate for the standard deviation, while in Ppk, the 'sample' estimate for the standard deviation is used.

Second Question:

I may be off base on this, but I thought that Ppk was used because the population standard deviation (sigma) is unknown? In this case, the biased calculation is used to normalize your results.

Anyone?
 

Kevin Mader

One of THE Original Covers!
Staff member
Admin
#8
Don,

Ppk. Never heard of it until I got into the automotive world. Ppk from my understanding is used to prequalify processes in early Control Planning/PAPP. I do not know enough about its derivation, so I can't dub it as complete "nonsense" although I feel in my gut that it is. My gut tells me that it is just another fine creation from the automakers to make life simpler and create useful shortcuts. What's the sense if it is scrutinized? Oh well.

Back to the group...
 
D

Don Winton

#9
Ppk from my understanding is used to prequalify processes in early Control Planning/PAPP.
I thought so as well. Personally, I do not see anything 'wrong' with a designation of Ppk using s rather than sigma. Other than what I stated earlier, that it is sometimes rare that sigma is actually known. But, that is too deep for here.

My gut tells me that it is just another fine creation from the automakers...
Yea, I would tend to agree with that.

Regards,
Don
 
D

Don Winton

#10
All this Ppk and Cpk stuff gives me a headache. Where the heck the concept of Ppk came from is beyond me. But Kevin, you raised an interesting point. I definitely do not envy you QS folks. Anyway:

Is the sample standard deviation (s, n-1) the same as the population standard deviation (sigma, n)?
No. When sigma is unknown, it is normally estimated. An unbiased estimator of sigma is given in Grant and Leavenworth, page 107 (too cumbersome to go into here). Statistically speaking, in a manufacturing environment, sigma is never known, and must always be estimated. Sorry, the statistical purist in me. For practical applications, s is usually a reliable estimate of sigma after 60 data are available. When less than 60 data are available, page 5 of Cpk.pdf (in the PDF Zone here at the Cove) gives ‘correction’ factors that also give an unbiased estimate of sigma. A derivation of these factors is also given at the end of Cpk.pdf for those interested.

For simplicity’s (please) sake, derive the unbiased estimate of sigma and use Cpk and avoid all this Ppk nonsense. It appears to be more trouble than it is worth. The unbiased estimate of sigma satisfies the ‘statistically valid’ requirement, so it should not be questioned.

Does this help, or did I just muddy the water.

Regards,
Don
 
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