# Converting from AQL to CPK

#### dac425

##### Registered
I recently had an interesting problem in acceptance criteria between 2 companies being different (AQL vs CPK). Acceptance criteria for a given lot was AQL at one company, and the other CPK.

The question was how to convert between the 2 and I found no straightforward answer. However, I thought because AQL sampling is prescribing a sample size for 0 defects you could determine the reliability with confidence (choosing 95%) using n= ln(0.05)/ln(reliability) of a sample.

I also thought I could make use of the k-factor for determining if variable data is meeting a confidence/reliability level. This is typically found in charts in my experience, but I did find the equations to calculate it. I thought this k-factor could be calculated for an equivalent confidence and reliability level to that calculated for AQL and sample size (The sample size that will be used to determine CPK, which is not necessarily the same as AQL sample size). The K-factor being the number of standard deviations the mean of the sample must maintain from the spec limit (2-sided in my case, but that same logic should apply for a 1-sided k-factor) can then be simply divided by 3 to determine the CPK.

Is this an accepted/known method? Would there be any issues with using this to justify equivalence?

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#### Semoi

##### Involved In Discussions
There exists many different methods to calculate "an AQL sampling plan" as well as for acceptance criteria using a Cpk value. Thus, answering your question is difficult, because details matter.

The "RQL sampling plan" you are describing seams to use a tolerance intervals for a proportion. It is commonly used if we have attributive data. In contrast, for continuous variables the Ppk-Method is commonly used. Again, the method which I prefer is based on calculating a tolerance interval. If you wish, you could use the expected failure rate to relate the RQL and Ppk method -- others will tell you that this emphasises a wrong perspective. Personally, I don't see why we should try to relate these two methods: Either of them has clear input parameters and they are typically used for different data types. Thus, I reckon it's pretty clear which one to use for a given characteristic.

However, I thought because AQL sampling is prescribing a sample size for 0 defects you could determine the reliability with confidence (choosing 95%) using n= ln(0.05)/ln(reliability) of a sample.
This is not true. We can calculate tolerance intervals for any given defects number -- and we should, because else the probability to fail the test might be unacceptably high.

The k-factor method is equivalent to the Ppk method -- if we consider tolerance intervals. The relationship is Ppk = k/3, as you stated. I believe this equivalence is mathematically exact for single sided specifications, and only approximate for two sided specification. In addition, the data must follow a normal distribution.

#### Steve Prevette

##### Deming Disciple
Super Moderator
I do not believe it to be appropriate to convert AQL to CPK. AQL is Acceptable Quality Level. "n ISO 2859-1, the AQL is defined as the “quality level that is the worst tolerable.” (from What is the AQL (Acceptance Quality Limit) in QC Inspections?) That is the "goal", an aspriation. It may not be reality. It is the voice of the cusotmer (a phrase from Don Wheeler). The CPK relates to is the goal being met (though better to use SPC to get the "voice of the process".

So a supplier could state - sure this is our AQL, while the actual acceptance rate is far from that number.

#### Bev D

##### Heretical Statistician
Super Moderator
First it is best to ask your Customer who is requiring Cpk as a release criteria what they want you to do. Steve is exactly right AQL and Cpk are NOT the same thing even if you take into account the difference between attribute data and continuous data. In addition Cpk is actually a tightening of whatever specification your Customer has provided, If the Cpk is greater than 1, since all observed results must be substantially within specification.

Another note: While this may be just a matter of semantics and/or sentence structure, it does matter for clarity of interpretation and understanding: AQL sampling does not proscribe a sample size for zero defects - there are many plans that have accept numbers (in the sample) that are greater than 0 and all AQL plans (and RQL plans for that matter) must have some stated defect rate for the lot even if the inspection sample has an accept number of 0.