Sample Size based on Process Capability Indices (Pp/Ppk)?

J

jasonb067

#11
Here is an interesting point. You can plug in all sorts of data into all sorts of equations, but there are some empirical statistical concepts that you need to consider. First what is the point of the question? Is the point to determine what your ongoing sample size should be if you know your Cpk (and...assuming your Cpk is even valid, which should always be called into question)?

If so, then the next empirical statistical point is that the lower the probability of defect (higher Cpk) the larger your sample size has to be. It reaches 100% very quickly. Why?The fewer rejects, the harder they will be to find - pure and simple. Think needle in a haystack. Would you assume the fewer needles you have, the less hay you would need to dig through to find them? Well, the same issue applies to sampling. So, seeking out any statistical method that would support lesser size sample when your capability is greater does not end up making any sense if you think of the point of the exercise.

What if the purpose of sample is not to detect but to demonstrate ongoing capability?
 
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B

Barbara B

#12
Thanks for the link, but OC-curves for acceptance sampling are different from what this thread is about. (The difference between OC-curves for acceptance sampling and hypothesis testing is given in detail i. e. in Montgomery "Introduction to statistical quality control", ISBN 978-0470169926, p. 604ff. for acceptance sampling and p. 124ff. for hypothesis testing.) If you're using fractions of nonconformance you're right about the issue with small numbers.

But if you take the numbers directly (using mean and standard deviation or some kind of ratio of both (i. e. Pp/Ppk), the sample size is not affected by the small-fraction-nonconformance-problem. The distribution of Pp and Ppk are based on the distribution of mean and standard deviation (Normal and Chi² for normally distributed data).

I'm searching for informations about sample size calculations based on Pp/Ppk itself (with respect to mean, standard deviation and their distributions). One advantage of such a formula would be that small-fraction-nonconforming-issues could be avoided :bigwave:
 

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#13
What if the purpose of sample is not to detect but to demonstrate ongoing capability?
Then the sample size needs to be large enough to capture the variations the process exhibits. Note that nearly all processes are multimodal, as illustrated by the total process variance equation - so a sample large enough to include the effect of all of the statistically significant variances is needed for the representation of the capability index to be valid. That depends on the process. It also shows why it is difficult to assure that a simple, single index value is meaningful over time - as it is designed to describe a snapshot in time (and not very well, either).
 
Last edited:

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#14
For the OC curves Barbara B is referring to related to the power curve, you may want to review this link. Still, to increase the power, you need to increase the sample. Sorry for the mix-up....but keep an eye on the other link too if you have to consider acceptance sampling.
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#15
One of my customers told me that one of his customers has shown him a paper which provides sample sizes depending on process capability indices (Pp/Ppk). With this method appropriate sample sizes for process validation could be picked out of a table (without further knowledge about mean, variation, tolerance limits, etc.)

Hopefully someone in the Cove could provide further informations about this "sample size based on Pp/Ppk"-method!
As a reminder to those in this thread I wanted to refocus us on the original request - Barbara is looking for a paper or discussion that could leverage Ppk values to determine sample sizes for process validation.

She is not looking for Acceptance sampling sample sizes or ongoing process capability...(unless I have misinterpreted - Barbara please correct me if I'm wrong)

Bob has kindly provided a reference link for those interested in Acceptance sampling.

For Jasonb067: on-going process capability can be assessed with an appropriate control chart. Once a baseline capability has been established, the control chart can monitor that the level is being maintained or that capability has improved or degraded with relatively small sample sizes. (as always with categorical data and small defect rates, sample sizes increase dramatically regardless of the statistical question) This is one reason why control charts and process capability are so intimately linked...
 
B

Barbara B

#16
As a reminder to those in this thread I wanted to refocus us on the original request - Barbara is looking for a paper or discussion that could leverage Ppk values to determine sample sizes for process validation.

She is not looking for Acceptance sampling sample sizes or ongoing process capability...(unless I have misinterpreted - Barbara please correct me if I'm wrong)
Thanks BevD, that's exactly what I'm searching for :)
 
A

aproddutoor

#17
I'm trying to update our Process Validation Procedure and I'm looking for the same information, a mathematical derivation of some sort to calculate variable sample sizes based on confidence and reliability to meet a certain level of Ppk . Does anyone have an answer to this?
 
B

Barbara B

#19
Nupp. Like Bev_D stated earlier, it seems to be a rumour that the sample size could be calculated for Pp/Ppk with respect to type I and type II error risk (alpha and beta resp.)

But if you find something, we'll appreciate the information ;)
 
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