# Relationship between Normal Distribution and AQL

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#### siumks

I have a question below:

What is the relation between normal distribution and AQL?
I know AQL is set up under 95% confidence level of normal distribution, but what is the theory behind it?
How can the confidence level link up with AQL?

Thanks so much and I appreciate your response Last edited by a moderator:

#### Paul Simpson

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I have a question below:

What is the relation between normal distribution and AQL?
So far, so easy. Normal distribution and AQL are largely unconnected. :mg:

I know AQL is set up under 95% confidence level of normal distribution, but what is the theory behind it?
How can the confidence level link up with AQL?

Thanks so much and I appreciate your response ... and this is where I go aainst what I have said above. If you take a sample to decide whether to accept or reject the batch you want to have confidence that the sample you take is an accurate representation of the batch as a whole.

So if you take your sample of a certain size based on the size of the whole batch there is a probability say 95% although there are others that the sample is representative of the whole population. Now this probability is based on the normal distribution - saying that if you took a large enough number of samples of the batch the results would tend towards a normal distribution of results for the attribute / measurement you were looking for.

Hope this helps.

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Now this probability is based on the normal distribution - saying that if you took a large enough number of samples of the batch the results would tend towards a normal distribution of results for the attribute / measurement you were looking for.
Boris, I would disagree with the statement that the probability is based on the normal distribution. AQL for attribute sampling plans are based on the BINOMIAL distribution. For a given sample size and observed defects, you have a 95% probability of accepting a sample from a process with an average percent defective of p (this is the AQL).

Assume we have a sample size of 30 units and accept on zero defects your AQL is 0.17%. In other words, we have a 95% confidence of accepting a sample of 30 units having zero defects if the process average defective is 0.17%

Does that make sense?

#### Paul Simpson

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I knew in advance I'd regret this! Boris, I would disagree with the statement that the probability is based on the normal distribution. AQL for attribute sampling plans are based on the BINOMIAL distribution. For a given sample size and observed defects, you have a 95% probability of accepting a sample from a process with an average percent defective of p (this is the AQL).

Assume we have a sample size of 30 units and accept on zero defects your AQL is 0.17%. In other words, we have a 95% confidence of accepting a sample of 30 units having zero defects if the process average defective is 0.17%

Does that make sense?
1. The OP didn't mention attribute sampling and neither did I
2. I understand the difference between attribute and variable sampling - the OP did not ask for AQL in relation to either
3. The 95% confidence figure applies to both the Gaussian (normal) distribution and to the binomial
4. I realize normal only ever approximates to binomial
5. My guess is the OP wanted a simple explanation

My brief explanation was to (unsuccessfully it seems) explain the link between AQL and confidence without going through the whole derivation.

Memo to self: 'Don't get involved in statistical discussions on the Cove - not even if your life depends on it! '

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
I knew in advance I'd regret this! 1. The OP didn't mention attribute sampling and neither did I
2. I understand the difference between attribute and variable sampling - the OP did not ask for AQL in relation to either
3. The 95% confidence figure applies to both the Gaussian (normal) distribution and to the binomial
4. I realize normal only ever approximates to binomial
5. My guess is the OP wanted a simple explanation

My brief explanation was to (unsuccessfully it seems) explain the link between AQL and confidence without going through the whole derivation.

Memo to self: 'Don't get involved in statistical discussions on the Cove - not even if your life depends on it! '
That is not true Boris...your insight and contribution is great. In the case of continuous data, AQL would be the 95% confidence level so you are correct. A

Neat discussion - thanks for helping link AQL and sampling.

I do have a question though - if the data is continuous, is correct that it's the repeated sampling from the lots or process and the use of the sample means that produces the normal/gaussian distribution via the central limit theorem? I seem to recall that being a bit of a hall pass for the base data not needing to be normally distributed to (correctly) use the confidence interval formula, which does have a normal depedency mathematically.

Thanks for any clarification!

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#### NumberCruncher

If you take several samples from a distribution (lets say the length of bolts off a production line). Take 5 bolts and take the average length. Record that number. Then take a different sample of 5 bolts and take the average length. Record that number.

Keep doing this say, 50 times, then plot a frequency histogram of the average values. You will get an approximately normal curve, even if the original distribution of bolt lengths is not normal.

Clearly, if the original, individual bolt lengths are normally distributed, the histogram of averages will become normal looking far faster than if you had a wildly skewed distribution of individual bolt lengths.

NC

A

NC -

Thanks. Yeah - it's the central limit theorem in practice. I was looking at the OP's question about how the AQL and the normal distribution are related and the answer is that they are not necessarily related directly, BUT are related through the use of repeated sample means, which as you note, do follow the normal distribution as the number of included sample means increases (sort of answers why you should use 30 for many samples).

Cheers

#### Jim Wynne

Staff member
NC -

Thanks. Yeah - it's the central limit theorem in practice. I was looking at the OP's question about how the AQL and the normal distribution are related and the answer is that they are not necessarily related directly, BUT are related through the use of repeated sample means, which as you note, do follow the normal distribution as the number of included sample means increases (sort of answers why you should use 30 for many samples).

Cheers
You should be aware that when "AQL" is invoked (as in ANSI/ASQ Z1.9 or Z1.4, e.g.) the assumption is sampling of lots (static populations) and not continuous data. AQL in sampling plans like these is not intended to be able to hit a moving target.

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#### siumks

Hi Steven,
if AQL is talking abt 95% probability, so could you share why it is 95%, but not 97% or 90% neither......
i still not understand....pls help