# Statistical basis for square-root sampling

#### Scott Catron

##### True Artisan
Super Moderator
Hi, I'm new. Found this place while googling for an answer to this question: Is there any statistical basis for square-root sampling?

Background: We've just started bottling topical pharmaceutical solutions for a new customer and we're dealing with larger numbers then we have in the past. Our current procedures use MIL-STD-105D for sampling and acceptance criteria. We sample and check the empty bottles as they come in and then the filled bottles after they're done. Some in production have asked about square-root sampling since it involves fewer bottles to check (example: for 210,000 bottles received, Mil-105 says check 800, square root would be 458). From the couple relevant things I found via Google is doesn't sound like there is a statistical rational for using square-root sampling, plus there's no discussion about acceptance/rejection numbers.

Any help or advice is appreciate.

Thanks, Scott

D

#### David Hartman

Scott_Catron said:
Hi, I'm new. Found this place while googling for an answer to this question: Is there any statistical basis for square-root sampling?

Background: We've just started bottling topical pharmaceutical solutions for a new customer and we're dealing with larger numbers then we have in the past. Our current procedures use MIL-STD-105D for sampling and acceptance criteria. We sample and check the empty bottles as they come in and then the filled bottles after they're done. Some in production have asked about square-root sampling since it involves fewer bottles to check (example: for 210,000 bottles received, Mil-105 says check 800, square root would be 458). From the couple relevant things I found via Google is doesn't sound like there is a statistical rational for using square-root sampling, plus there's no discussion about acceptance/rejection numbers.

Any help or advice is appreciate.

Thanks, Scott

Scott,

I'm not familiar with square-root sampling, but have you considered C=0 sampling (it too reduces the sample size over MIL-STD-105D)?

Some information on C=0 can be found by doing a search routine on this site.

#### CarolX

Trusted Information Resource
Re: Re: Statistical basis for square-root sampling

ddhartma said:
Scott,

I'm not familiar with square-root sampling, but have you considered C=0 sampling (it too reduces the sample size over MIL-STD-105D)?

Some information on C=0 can be found by doing a search routine on this site.
David,

My understanding of C=0 sampling was you used the standard sampling called for in MIL-STD-105, but you change your accept/reject numbers to 0/1.

Scott,
Look through MIL-STD-105 at the special sampling plans. They are designed for taking smaller samples than the "standard" levels. Review the OC curves to analyze you consumer and producers risks and see what works for you.
And, by the way
Welcome to the Cove!!!!

CarolX

R

#### ralphsulser

Re: Re: Re: Statistical basis for square-root sampling

I recall a provision in MIL-STD-105 allowing for reduced sampling if 10 consecutive lots have been accepted. However, if one lot is found to be rejectable, the you must revert to normal sample sizes untill you reach the 10 consecutive lot acceptable again. Of course this assumes C=0
Look at you sample tables and footnotes. It's been a while since I used them.

#### howste

##### Thaumaturge
Trusted Information Resource
Re: Re: Re: Statistical basis for square-root sampling

Hey, another Utahn! Welcome!

I'm not familiar with square-root sampling either. I think CarolX hit the nail on the head though. Look at the OC curve for whatever sampling plan you use to determine what the risks are. If they're acceptable to you and your customers, go for it.

A

#### Atul Khandekar

I am not familiar with Sqare root sampling..but just wondering if this has anything to do with the fact that Poisson distribution for counts can be converted to Normal using square root transform. (I may be wrong!)

Also: (broken link removed)

R

#### Rick Goodson

Scott,

Welcome to the Cove.

I also have not run into anything called square root sampling, so I can not help with that. However I think you can quickly see the problem with that type of plan if you look at what happens to the OC curve for a % based plan. Plot an OC curve for say 10% plans using lot sizes of 90, 300, and 900 with a c=0 acceptance number. As the lot size increases the curve gets tighter or conversely as the lot size decreases the curve gets looser and has a substantially higher probability of accepting product with a higher percent defective than you want. I suspect you will find the same thing with a square root plan. The basic problem is that an acceptance sampling plan should deliver close to the same protection (read similar OC curve) regardless of the lot size.

As mentioned earlier, consider c=0 sampling plans. Look for a publication by Nicholas Squeglia called "Zero Acceptance Number Sampling Plans". Be carefull when you look at these. While there is a significant reeduction in sample size, and the plans are statistically sound, the OC curve on a c=0 plan is not an "s" shape, it is concave. (By the way, in MIL-STD-105 all of the 0 acceptance numbers also have concave OC curves) You have an extremely long tail to the right. In essence there can be a significant probability of acceptance of a lot of material that has somewhat worse quality than you want. Remember with most "s" curve plans the dropoff is rather steep after the 90% acceptance point. Not so with c=0 plans. Make sure you look at the OC curves for these plans.

Good luck.

#### Scott Catron

##### True Artisan
Super Moderator
Re: Re: Re: Re: Statistical basis for square-root sampling

Thanks for the replies and welcomes:

ralphsulser said:
I recall a provision in MIL-STD-105 allowing for reduced sampling if 10 consecutive lots have been accepted. However, if one lot is found to be rejectable, the you must revert to normal sample sizes untill you reach the 10 consecutive lot acceptable again. Of course this assumes C=0
Look at you sample tables and footnotes. It's been a while since I used them.

Paging through Juran's today I happened upon the same thing - it specifies reduced sampling if 5 consecutive lots are accepted.

I'm not sure C=0 sampling will work for this process. We have 17 different criteria we check on finished lots, with AQLs ranging from 0 to 6.5.

I'm not familiar with OC curves but have found the pertinent section in Juran's.

I guess the answer to the original question is no, there's no (implicit) statistical basis for a square-root sampling routine. I suppose if the OC analysis could justify it we could use it.

(Just to note - I haven't had any formal training in these techniques - just a stats class in grad school for the physical sciences)

Thanks for the help,

Scott

D

#### David Hartman

Scott_Catron said:
Hi, I'm new. Found this place while googling for an answer to this question: Is there any statistical basis for square-root sampling?

Background: We've just started bottling topical pharmaceutical solutions for a new customer and we're dealing with larger numbers then we have in the past. Our current procedures use MIL-STD-105D for sampling and acceptance criteria. We sample and check the empty bottles as they come in and then the filled bottles after they're done. Some in production have asked about square-root sampling since it involves fewer bottles to check (example: for 210,000 bottles received, Mil-105 says check 800, square root would be 458). From the couple relevant things I found via Google is doesn't sound like there is a statistical rational for using square-root sampling, plus there's no discussion about acceptance/rejection numbers.

Any help or advice is appreciate.

Thanks, Scott

Scott, First please forgive me for not giving you a hearty "Welcome to the Cove." in my first response.

Secondly, I did a search using Dogpile.com and came up with the following webpage which seems to provide some clarification as to where the square-root sampling method came from and some of the problems with it.

Hope this helps.

R

#### Rick Goodson

Hi Scott,

Your last post raised an point I want to address, OC Curves. The basis of an sampling plan is the risk associated with using it, both to the buyer and to the seller. The Operating Characteristic curve defines those risks for any sampling plan. I would highly suggest you find a book on sampling plans, or on quality control, and read up on OC curves. Using a sampling plan without understanding the risks is very not a sound idea.