# Sampling Plan to determine sample size/frequency based on a Given Rate of Production

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#### bimco

I'm looking for a sampling plan/table to determine sample size and inspection frequency based on a given rate of production. For example, a process produces 500 pieces. How many pieces should be checked and how frequently?

#### Marc

##### Fully vaccinated are you?
If you're looking for an AQL (Acceptable Quality Level), why not use the old Mil-Std-105? It's technically a canceled document (as is the way with many old military specs) but you can still use it. I know an organization 'took it over' but I can't remember who it was.

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#### Don Winton

bimco,
I agree with Marc. Even tho MIL STD 105 is discontinued, the OC's and sampling plans are still statistically valid. Use these as a good guide. I recently received a post that gave guidance as to sample sizes:

n>=((Za/2*sigma)/E)^2 wher Za/2 is (1-confidence(i.e.0.95))/2. Sigma is standard deviation of the data and E is the expected Error. This is also considered the difference between the mean and your sample and the mean (estimated, maybe) of your population.

I have another equation somewhere (?) and will also try to forward this one as well.

Regards,
Don

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#### Dawn

Sampling Plans

Wanna talk sampling plans? I know. I read and read and I just don't get it. Wish I was born with a statistics mind. How do you determine the OC curve in the first place? I don't see the sample size anywhere on the OC curve samples I have looked at and I don't see any explanation on how they get it on there. Lost again....

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#### Batman

Dawn, I cannot answer you specifically, since I am also continually puzzled by statistics beyond Xbar and sigma, and histograms. However, Stan Hilliard has a sampling plan website called www.samplingplans.com. He also sells software there. I have never seen the software and am not promoting it, but he does have some info there.

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#### Don Winton

Dawn,

Oops. Forgot something. I detailed the distributions to use, but forgot to describe when to use them. Here we go:

Binomial Probability Distribution: Applies when the population is large (greater than 50) and the sample size is small compared to the population, generally less than 10%.

Poisson Probability Distribution: Is an approximation to the binomial when P is equal to or less than 0.1 and the sample size is large. (I prefer this one).

Hypergeometric Probability Distribution: Whenever the sample is drawn from a smaller, finite lot. This calculation is a bear, but spreadsheets and computers make it easier these days.

Then, there is always the normal distribution. It can be used as a probability distribution as well when the population size is large.

The above are general guidelines. There are no firm, fixed rules when to use one or the other. My suggestion would be to balance time versus benefit in the calculations.

Generally, the sample tables in MIL-STD-105 are statistically valid, even though the document itself is obsolete. Rather than design sampling plans from scratch, I suggest, if a sampling plan is used, design it from these tables.

Regards,
Don

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#### Don Winton

Wanna talk sampling plans?

I personally am not a big fan of sampling plans, but that is another story.

How do you determine the OC curve in the first place?

The OC curve is developed by determining the probability of acceptance for several values of incoming quality. Pa is the probability that the number of nonconforming in the sample is equal to or less than the acceptance number for the sampling plan.

There are three distributions that can be used to find the probability of acceptance: the hypergeometic, binomial and the Poisson distribution. The Poisson distribution is the preferred because of the ease of Poisson table use. Be sure you can satisfy the assumptions for Poisson use. The Poisson formula is:

P = [(e^-np)*((np)^r)]/r!

n = Sample Size
p = Proportion nonconforming
r = Number of nonconforming or less

Assume n = 150 and r = 3, (called c in sampling plans) then:

Lot Percent Nonconforming
1%, np = (150)(0.01) = 1.50, P(r <= 3) = 0.93
2%, np = (150)(0.02) = 3.00, P(r <= 3) = 0.65
3%, np = (150)(0.03) = 4.50, P(r <= 3) = 0.34
4%, np = (150)(0.04) = 6.00, P(r <= 3) = 0.15
5%, np = (150)(0.05) = 7.50, P(r <= 3) = 0.06
6%, np = (150)(0.06) = 9.00, P(r <= 3) = 0.02

You then construct the table from these values with Lot Percent Nonconforming as the X value and Pa as the Y value.

Hope this helps.

Regards,
Don

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#### Woraphot

According to QS-9000 3rd edition Element 4.10 Inspection and Testing "4.10.1.1 Acceptance criteria for attribute data sampling plans shall be zero defects........"
Does it mean that Mil-std-105 can not meet to QS-9000 requirements on this item? Because of it allows acceptance not zero defects depend on AQL level and Lot/Batch size.
Thanks,

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#### Don Winton

Woraphot,

I do not do the QS thing, but I think what they are talking about might be this. The tables contained in MIL-STD-105 contain Ac and Re codes. If you use one of the tables from this standard, you must use an Ac of zero and an Re of one for the plan. Perhaps some QS folks could expound on this.

Regards,
Don

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#### Batman

I believe 4.10.1.1 is refering to the acceptance criteria, not the sampling itself. The attribute gage (in this case, a "hard" gage) must not allow any defects to pass. Your sampling plan - size, frequency - is up to you, as long as it is adequate. Any other attributes - color, flash, chips, for example, that are visually inspected must have "border" examples agreed to by your customer