Sampling plan to detect defective units - Given N=6000, PPM=500

K

keatwoon

#1
Can anyone help to answer the following question:

Given N=6000, PPM=500;
How many sample size do I need, to be 95%confidence, to detect and prevent the use of the unacceptable materials in the given lot size? Thanks.
 
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Bev D

Heretical Statistician
Staff member
Super Moderator
#2
keatwoon said:
Can anyone help to answer the following question:

Given N=6000, PPM=500;
How many sample size do I need, to be 95%confidence, to detect and prevent the use of the unacceptable materials in the given lot size? Thanks.
given that this isn't a homework question:
I would use the Poisson and the sample size to determine if the lot is not 500ppm or worse is 5992. sorry. such small defect rates require large sample sizes when you are using only pass/fail criteria. If you were to be measuring for continuous data characteristics, the sample size could be MUCH smaller.

the formula for the Poisson reduces to the following when we accept on 0, reject on 1: -ln(1-confidence)/p if 500 ppm p = .0005

so -LN(1-.95)/.0005 = 5992
 

Tim Folkerts

Super Moderator
#4
Yes, when the sample size gets close to the lot size, then the hypergeometric distribution is a better at getting the correct probabilities. Unfortunately, the equation is somewhat complicated and involves factorials. Excel does have this functionbuilt in, but factorials for large numbers become huge! Sample sizes much over 100 return an error.

Details about the eqaution can be found at
http://mathworld.wolfram.com/HypergeometricDistribution.html


Part of the challenge is that your error rate is so small. 500 ppm = 1 part per 2000 = 3 parts per 6000. Assuming 500 ppm is the limit on acceptable lots, you are trying to deterine if there are 4 or more defects out of the 6000 pieces.

If you have to use a pass/fail type test and if the test is labor intensive, this might be a good place to consider a continuous sampling plan. I haven't work the exact numbers, but it would be something like:
* reject the lot if 1 of 500 is bad
* reject the lot if 2 of 1500 are bad
* reject the lot if 3 of 3000 are bad
* reject the lot if 4 out of any number are bad
* accept the lot if 0 of 2000 is bad
* accept the lot if 1 of 4000 is bad
* accept the lot if 2 of 5000 are bad
* accept the lot if 3 of 5900 are bad

Always note that this assumes a random sample. you shouldn't just take the samples in order because the defects are quite likely to be clustered in many real situations.

Tim F
 

Helmut Jilling

Auditor / Consultant
#5
Tim Folkerts said:
Part of the challenge is that your error rate is so small. 500 ppm = 1 part per 2000 = 3 parts per 6000. Assuming 500 ppm is the limit on acceptable lots, you are trying to deterine if there are 4 or more defects out of the 6000 pieces.

If you have to use a pass/fail type test and if the test is labor intensive, this might be a good place to consider a continuous sampling plan. I haven't work the exact numbers, but it would be something like:
* reject the lot if 1 of 500 is bad
* reject the lot if 2 of 1500 are bad
* reject the lot if 3 of 3000 are bad
* reject the lot if 4 out of any number are bad
* accept the lot if 0 of 2000 is bad
* accept the lot if 1 of 4000 is bad
* accept the lot if 2 of 5000 are bad
* accept the lot if 3 of 5900 are bad

RE: PPM=500 - if this is internal PPM, it is indeed pretty good. If 500 is an external PPM performance, it is not good at all. An external 500ppm would fall below the performance of almost every certified client I work with.

Also, please bear in mind, you are discussing AQL levels. If you are talking about automotive clients, the default is usually C=0.
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#6
Sorry - but in this case, your best approach is either a poke yoke inspection device and 100% inspection (if possible) OR use a "variables" sampling plan if your characteristic of interest can be measured and expressed with continuous data (as opposed to pass/fail categorical data). A sample plan that uses continuous data will enable significantly smaller sample sizes...
 

Tim Folkerts

Super Moderator
#7
Bev,

I have to agree with you. :agree1:

I was answering the statistics question of what sort of plan would be required to meet the original "specs" of a sampling plan to acheive a particular confidence level, etc. It turns out that a sampling plan to acheive the desired goals as stated is not particularly practical. Even the continuous sampling plan would require testing a good fraction of the parts from each lot.

As you rightly point out, this is a case where the original problem should re-thought. You provide two excellent alternatives - 100% testing or finding a variable to measure rather than a pass/fail test.


Tim F
 

Statistical Steven

Statistician
Staff member
Super Moderator
#8
Bev D said:
given that this isn't a homework question:
I would use the Poisson and the sample size to determine if the lot is not 500ppm or worse is 5992. sorry. such small defect rates require large sample sizes when you are using only pass/fail criteria. If you were to be measuring for continuous data characteristics, the sample size could be MUCH smaller.

the formula for the Poisson reduces to the following when we accept on 0, reject on 1: -ln(1-confidence)/p if 500 ppm p = .0005

so -LN(1-.95)/.0005 = 5992
Excellent post! Just a caveat, Poisson should be used when npq <5 otherwise use the binomial. In this case, the difference is in the 3rd decimal place.
 
N

nielsmx

#9
Isn't that 100% inspection?

Sorry, I'm not a statistician, but if I see the number 5992 out of 6000, I get the feeling that we're talking about 100% inspection. And once we are talking about 100% inspection, we can throw away the defective pieces and make sure that we deliver 6000 pieces defect free.
What we still have to do is checking our inspection for defects. Too often the production (or development) process is thought to be in need of inspection, however, I assume that inspectors are also human and as much prone to producing defects, so if we did 100% inspection, what will be our confidence of delivering according to the spec?
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#10
Statistical Steven said:
Excellent post! Just a caveat, Poisson should be used when npq <5 otherwise use the binomial. In this case, the difference is in the 3rd decimal place.
yes the theoretical requirement for the Poisson is not met. This is a consistant problem with very low defect rates...but we must remember that both teh Poisson and the Binomial are only models and estimates. The real proof is if they are useful for need. (to paraphrase George Box). Other factors suchas how the samples are taken and how the defects may or may not cluster will add more 'inaccuracy' to the result thant the choice of distributional model. and as you point out the differences are slight anyway.
I choose to use the Poisson in most cases (not all) because it is easy to calculate

But of course the real point here is that with small defect rates we need to stop looking for proportional sampling plans and begin assessing alternatives sucha s poke yoke and continuous data plans. you simply cannot guarantee zero defects with proportional sampling plans adn this is where so much of industry is headed - an drightly so in my opinion.
 
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