# AQL, Lot Size vs Sampling Plans

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#### unueco

Greetings!

I'm am looking for a bit of technical/theoretical background concerning acceptance sampling plans.

Almost from my first day on the job working in QA, I've had various "gurus" in the field tell me not to use any of the standard sampling plans (ISO 2859-1, ANSI/ASQ Z1.4, mil std 1916, etc etc). No instead the true QA practitioner should develop his/her own specific sampling plan based on desired probabilities of acceptance based on fractional lot defective using something like a Larson Nomograph.

I thought I understood. But I'm finding the more I read on this, the more confused I am getting.

For example:
Most standard plans have "Lot size" as an input parameter. But its is not a parameter for Larson's. Is this because Larson's is based on the use of a binomial distribution -- which itself is based on an assumption of an infinite lot size? If so, then doesn't the accuracy of this approach depend on having "large" lots sizes? Is one better off using the standard tables for smaller lot sizes? And just how big is big anyway??

Conversely, Mil-std 1916 specifically states that it is designed to be used *in lieu of* "AQL based sampling plans". How does that work?

So...as I said...I'm a bit confused.

Any assistance, guidance, or pointers to reference sources is highly appreciated!

Thanks!

#### Statistical Steven

##### Statistician
Super Moderator
A few points to keep in mind. Most sampling plans assume infinite lot sizes and use the binomial to estimate the AQL and LTPD. The problem with Z1.4 and other standards, they are AQL based. Without the LQ, it is hard to evaluate a sampling plan. Not Mil-Std 1916 gets aways from AQL based plans, but using the binomial directly, you can calculate an OC curve for ANY sampling plan.

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#### unueco

Hi, thank you for your response!

Without the LQ, it is hard to evaluate a sampling plan.

Can you please explain what you mean by LQ?

Not Mil-Std 1916 gets aways from AQL based plans, but using the binomial directly, you can calculate an OC curve for ANY sampling plan.

I also am not sure I follow you here. I thought you need the AQL to construct a sampling plan via binomial distribution and OC curves.

So, I am curious how Mil-std 1916 was developed.

Also, I'm curious if the fact that sampling plans based on the binomial dist are based on the assumption of "large" lot size would affect the accuracy of their usage as opposed to plans that include lot size as a parameter.

Thank you!

#### David-D

##### Involved In Discussions
One of the best ways to compare various sampling plans (whatever their origin) is based upon the Operating Characteristics (OC) Curves. It allows comparison of probability of a?ceptance (Pa) over the range of quality (fraction or % defective). Often a couple of specific points are compared between plans (or used to generate them by curve fitting): AQL ("acceptable quality level", the % defective which would give 95% probability of acceptance), LTPD ("lot tolerance percent defective", the 90% reject level (10% accept level)), and occasionally the indifference point (50/50 accept/reject). You can select the various points you want for the effectiveness of the sampling plan (based upon supplier and consumer risk) and then compare the ability of various sampling plans to curve fit to them.

Although I usually take the easy approach and use the binomial distribution to make my OC curves based upon ease of creation in excel and that generally my sample sizes are "small" compared to my lot sizes, you could do it statistically "correctly" to reflect the change in the defect % in the population based upon the previous samples pulled and use Poisson(?)(I might be wrong on the specific distribution, I don't have a book on my living room couch). Either way, in an truely AQL based plan (MIL-STD-105, ANSI Z1.4, etc) changing of the sample size based upon the code letter (where you move vertically within a column for a specific AQL) is going to have a much greater effect on the curve making it more descriminating (more like a step function) than whatever effect is related to sample size vs lot size.

David

#### David-D

##### Involved In Discussions
Can you please explain what you mean by LQ?

LQ is the Limiting Quality, also known as Lot Tolerance Percent Defective (LTPD) or sometime Rejectable Quality Level (RQL). It represents an unacceptable level of quality in a product that you want you're sampling plan to reliably reject. It protects/represents consumer risk (as the AQL does for supplier risk). It is often (by convention) the level of defects at or above which there is 10% or less probability of acceptance as the AQL is the defect level above which there is a 95% probability of acceptance.

David

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#### unueco

Thank you again for your response.

One of the best ways to compare various sampling plans (whatever their origin) is based upon the Operating Characteristics (OC) Curves.

So how would one go about doing that to compare say Mil Std 1916 to ISO 2859-1?

For that matter, how were the Mil Std 1916 compiled without use of an AQL? (which I thought comprised two of the four primary inputs to the creation of a sampling plan)

#### David-D

##### Involved In Discussions
MIL--STD-1916 is really a philosophical sampling plan rather than a statistical sampling plan. It was developed in the mid nineties with a focus on conforming product, corrective and preventive action, continuous improvement, and acceptance on zero defects. In contrast, the AQL based sampling plans deriving from MIL-STD-105 are statistically based - as long as the defect rate is low enough, the product will consistently. Within a specific AQL, the sampling plans are generally "matched" to have a similar likelihood of acceptance (~95%) for the specific AQL level and being more descrimnating as the sample and lot size increases. The sampling plans of 1916 are not as "matched" but generally are more rigorous as the verification level increases and as the lot size increases.

All that being said, you can easily compare the relative merits of different sampling plans (whatever the source) using the OC Curves. I personally find it easiest if you overlay them so that you can compare the relaticlve probability of acceptance (Pa) for similar levels of defects.

MIL-HDBK-1916, the guidance document for MIL-STD-1916, includes the OC curves for its various sampling plans as well as some of the rationale behind it; similarly MIL-STD-105 included its OC curves (most of which are duplicated in its successors). You can get a copy of both at:
https://www.assistdocs.com

David

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#### unueco

Hi and again thank you for your response.

I did check out the Mil-hdbk that you referenced. It seems to provide a lot of background information about QA methodologies etc. But, as for the OC curves, it seems that they only provide a curve for the Verification Level T of the Mil-std. And I'm not sure *how* they even do that one. I'll need to read the handbook more closely to get a better understanding.

Then I need to figure out how to do the same for ISO 2859-1. Any idea if there is a similar "handbook" for that standard?

Thank you!

#### David-D

##### Involved In Discussions
I don't have any experience with ISO 2859-1 but from the snapshots of the tables I saw on the web it looks like it is another duplicate of MIL-STD-105 (like ANSI Z1.4). I don't know of any readily available sources on the derivation of 105 but it does include the OC curves in it and they should be equally applicable to any other standard with identical sampling plans.

David