What is the Cpk for Zero Defects

[Lpitt56]

I monitor my specific processes in PPM. I have a customer that says thats ok but wants to see it converted to Cpk index. I have found several conversion tables that work nicely (simply find the PPM and next column show Cpk). However, I have a few processes that over 3 month (my tracking intervals) have no defects and therefore a PPM of 0.

My conversion charts stops at a Cpk of 2 which is equivilant to .002 PPM. My question.... is it appropriate for me to show a PPM=0 as a Cpk of 2.... or what would the Cpk be fore zero defects?

Thanks!

[Stijloor]

A Quick Bump!

Can someone help?

Thank you very much!!

[Barbara B]

Let me first say something about Ppk/Cpk to ppm conversion:

You can't convert Ppk/Cpk into ppm and vice versa.

Even if all the assumptions for Ppk/Cpk and ppm calculation are met (stability, normality and so on) you have to make additional assumptions to get a Ppk/Cpk out of a ppm (and the other way round). This has to do with the different formulas for Ppk/Cpk and ppm (see attached pdf).

Without further assumptions only a range for ppm out of Ppk/Cpk (or a range for Ppk/Cpk out of ppm) could be derived, depending on where the out of specification values occur (see attached Excel-sheet).

There are several webpages and other sources which provide a calculator or conversion tables, but these will only work properly under the additional assumption (like one-sided tolerances or a process mean in the middle of the tolerance for two-sided tolerances).

Back to your original question: Without the values itself it gets tricky to convert a ppm rate of 0 into a Ppk/Cpk-value. One solution is to assume that the next product will be out of the specification limit(s) and to calculate the worst-case Ppk/Cpk-value under these conditions.

E.g. if you have 100'000 products tested and all are within the specification limit(s), your ppm is 0. The worst-case assumption is that product #100'001 is outside the specification limit, leading to a maximal ppm rate of
1/100'001 *1'000'000 = 9,9999ppm
and a range for the Ppk/Cpk from 1.42 to 1.47.

[howste]

I agree with Barbara, that you can't really directly convert PPM into Cpk without using the actual data for the calculation. At best you can come up with an approximate value using Z-value tables.

There's a BIG problem with calculating a Cpk for 0 PPM though: A normal curve has tails that are asymptotic, meaning the tails approach zero but never reach it. Statistically speaking the Cpk value for zero PPM is infinity. I'd like to see the look on your customer's face when you tell them your Cpk = ∞.

[Bev D]

OK I agree completely Barbara and Howste. Converting a ppm to a Cpk is as ludicrous as converting a Cpk to a ppm. However, at the risk of perpetuating this ill-conceived abomination of statistical alchemy, sometimes it is best to pick your battles with your customers and this battle probably isn't worth it for this guy. His customer is probably happy if he can just check off the old Cpk box and there is a way to do that.

Several approaches can work (although they are all disconnected from reality, they are theoretically OK from a mathematical standpoint.)



Bayesian: This method requires that all units are inspected or tested.

P(d) = [d+1]/[n+2], P9d) is a defect rate adn can then be converted to ppm. d = number of defects and n = sample size; since d=0, the formula becomes P(d) = 1/[n+2].

Exact Upper Confidence Limit: A second method is to use the exact confidence interval for the Binomial distribution for d=0 and your sample size. Calculate the upper confidence limit (you can probably get away with a 95% confidence interval) and then take the defect rate represented by that upper limit, convert to ppm then to Cpk...
the formula in EXCEL is: BETAINV(1-alpha/2,d+1,n-d+1)
for d=0 and alppha = .05, this becomes: BETAINV(.975,1,n+1)


The Rule of 3s: A very simple approach to calculating the Upper Confidence Limit for zero defects in a sample is given by the formula:

p = 3/n for an alpha risk of .05

This formula is derived by solving the Binomial for the probability of seeing zero defects (P(0)=alpha) = (1-p)^n
This approximation works best for n> 20

*reference: “If Nothing Goes Wrong, Is Everything All Right?”, James A. Hanley and Abby Lippman-Hand, The Journal of the American Medical Association, April, 1983, Vol. 249, No. 13, pp1,743-1,745