# How to calculate the Probability of Deficiency of the AQL of an Inspection

L

#### leturc

Hi All,

I'm trying to calculate the probability of deficiency of an inspection conducted. We already know that AQL has 2 risks from it's nature:

• Good lots can be rejected, or
• bad lots can be accepted.
My qyestion is how we can define the probability of failure of a "accepted" inspection and vice versa.

#### Mike S.

##### Happy to be Alive
Trusted Information Resource
Try looking at the OC (operating characteristic) curves and/or tables for the sampling plan you are using.

For example, for a lot size of 50 and a sample size of 13, with no rejects allowed in the sample, the probability of lot acceptance is 50% if around 5% of the parts in the lot are bad.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
My question is how we can define the probability of failure of a "accepted" inspection and vice versa.
I'm not sure of your exact question. It would be helpful to understand what you mean by "failure" of an "Accepted" lot. Is it a lot having more defects than the AQL level? Is it a lot having more defects than the RQL or LTPD level? or is it the probability of the lot being rejected under the same plan later in time (such as at your Customer). Also what do you mean by the 'failure' of a rejected lot?

L

#### leturc

Hi all,

Thanks for your answers. In fact I'm working for a 3rd party inspection company and give inspection services for our customers. Often, our client come back to us with a complaint that they didn't found the same defects percentage and / or defects of our sampling. I explain them that our sampling is random and due to the nature of the AQL sampling there is always a risk to accept bad lots or refuse good lots but unfortunately they do not want to understand. So, I was wondering if it is possible to give them after each service the probability of failure of our sampling? I now that ISO 2859 gives us a table with some % but I'ld like to calculate the probability for each individual sampling. Is it possible?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
The dilemma is that the probability is based on the actual defect rate of the lot. Which you cannot know from sampling. The probabilities are based on what would happen if you sampled the same lot many, many times. It is accurate on average over the long run...it is not accurate for any individual lot. The best you can do is to share the OC curve of the sampling plan you use. That describes the range of probabilities associated with various actual defect rates...

Some people also forget that the AQL is the acceptable defect rate that will be accepted 95% of the time...they think it is the defect rate that will be rejected 95% of the time...

The worst part is that a non-homogenous distribution of defects in the lot cannot be overcome by random sampling. So non-homogeneity increases the actual probability of 'missing' a 'bad' lot.

The other factor is the measurement error which is also rarely accounted for. Every measurement system, including visual inspection, has measurement error. The OC curves 'assume' that no measurement error exists...

Simply put, sampling is imperfect. Improving the quality so that defects are very rare is the only reliable way of knowing if you have an acceptable lot or not...

#### Mike S.

##### Happy to be Alive
Trusted Information Resource
The worst part is that a non-homogenous distribution of defects in the lot cannot be overcome by random sampling. So non-homogeneity increases the actual probability of 'missing' a 'bad' lot.
If you homogenize the lot it can......

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
If you homogenize the lot it can......
homogenizing the lot is rarely possible across industries as it involves mechanical mixing like shuffling a deck of cards. some things in some industries can do this.

Donald Wheeler touches on this in his column "The Truth About Acceptance Sampling", Part 1, Quality Digest July 2014.
The Truth About Acceptance Sampling, Part 2 the following month is also interesting regarding the dichotomy of Acceptance Sampling...

#### Mike S.

##### Happy to be Alive
Trusted Information Resource
Yup, I'm a big Wheeler fan.

I agree you can't always homogenize a lot for sampling, but sometimes you can. The nit I picked was that you said you "cannot" overcome a non-homogenous distribution of defects in the lot, when sometimes you can.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
ah, so the nit I'll pick back is that I did say you can't overcome non-homogeneity with random sampling....

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
ah, so the nit I'll pick back is that I did say you can't overcome non-homogeneity with random sampling....
I will up the nit-picking and say you overcome non-homogeneity by doing stratified random sampling!

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