Specification Conversion from Sets to Individual Measurements?

T

the_joker

#1
I'm currently contracting a company and have run across a situation that I'm unfamiliar with. So, I turn to you all for help.

We are sampling for weight of a product and using a “weight per 10” (wt/10) value for inspection. It’s unclear why this practice was first started, but that’s not the issue right now.

I’ve currently been given reams of test data in a “weight per 1” (or each). I need to compare the statistics on the wt/1 samples as part of my evaluation of the test samples. To do this however, I need to convert the existing wt/10 specification to a wt/1 specification. Obviously, I can’t just divide by 10 as the probability distribution of the 10 units will come into play when getting the average which would be read as the 10 unit group. I believe this can be solved using the central limit theorem, but when I was looking at it I’m not directly seeing how to apply it into a simple equation for conversion.

For discussion purposes, let’s say the wt/10 specification is:
Min: 190
Nom: 195
Max:200

Making the theoretical target for each unit being 19.5. What would be the spec range for the individual parts?

Any help would be most appreciated.
 
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BradM

Staff member
Admin
#3
Now I'm confused. :DIf you are taking 10 readings and simply take the average of those 10, why can't you compare the 10 average to the one?

Are you wanting to see if the "1" is significantly different than the "10"? Is that your goal?
 
T

the_joker

#4
Ya, maybe I wasn't so clear. I'll try to elaborate.

We used some basic assumptions to convert the wt/10 spec to an each spec (using the central limit theorem and an assumption regarding the standard deviation of the overall population). You have to do this as saying "well the wt per 10 is 190 - 200 so the each is 19 - 20" is mathematically wrong. The probability distribution of the "group" widens the "each" spec.

I'm doing a specification validation / review. Data was taken on an each bais to be evaluated for interactions. However, when evaluating the individual data it was found that if a single unit becomes "too low" on this per 10 spec, we get an out of spec condition on a related dimension. So, we are now realized that we need to be concerned about individual outliers. My intent is to do something with the low end tolerance of the group spec to try to "reduce" the spread of individuals (or shift it). Thereby reducing / eliminating these low end outliers.

And to head off the obvious solution of changing to an "each" spec, well let's just say I was politely (or not so much) told "NO" in resounding capital letters by engineering.

So, I tryed to use a similar equation (basically the inverted solution getting me to the individual specs) to convert back from a unit spec to a tighter "group" spec. But the math is telling me I've got to shift the range WAY too far (like the new nominal is 206 and min is 201 which shifts us way over what we make now). So I believe my math and/or assumptions are wrong.

I am hoping someone else has dealt with this kind of problem in the past and can either give me an example of how to do it with more accuracy or advise a course of action regarding how to ether restrict the group spec or shift it to get the results I want. Basically I have a new low end spec for an individual part, but can't change the group spec to an individual.

The_Joker
 

Tim Folkerts

Super Moderator
#5
Odd... I thought I replied to this a day or two ago.

The simplest solution I can come up with is to take the "reams" of data you have for individuals and group them into sets of ten. Then use the old specs for the grouped data.


Converting the data to a "weight per one" was more problematic than I initially would thought. Using your example, the obvious value for the nominal would be 19.5 for one piece. However, dividing the limits by 10 (+/- 0.5) is too tight. From the Central Limit Thm, you could try expanding the limits -- in this case since the sample size was decreasing by 10, the limits would increase by 10^0.5. Thus you would get limits of +/- 1.6 instead of +/- 5 -- or 17.9 to 21.0 for the individuals

But there is a problem.

  • A bunch of individuals distributed within 19.5 +/- 1.6 would produce groups of ten distributed within 195 +/- 5 (which would be fine).
  • BUT a whole set of individuals distributed from 20.0 to 21.1 would fit these limits, but they would produce sets of 10 that had a mean of well over 200.
You can't legitimately work backwards from the central limit theorem in this way. The Theorem applies to data, not to spec limits.



Tim F
 
T

the_joker

#6
Thanks Tim,

I get what you are saying, but since you started down the path, I'll use more numbers to get at the crux of my problem.

So, as you mentioned I've now got a range of 19.5 +/- 1.6 to work with (17.9-21.1). This isn't really a spec. It's just the range of values a normal distribution would fall in given that I meet the group spec. I take all my data and let's just say it's all between 18.2-19.9 for numbers to work with. I also identify that the individual units at 18.2-18.5 have a subsequent defect due to their individually low value - holes, thin spots, bad shot, whatever. I'd like to now "modify" my 17.9-20.1 "spec range" (really the probability not a spec) and truncate everything below 18.5. So, I now have a chopped distribution that is 18.5-20.1 dumping one of the tails.

I'm looking for the math to take my fancy new 18.5-21.1 probability limits into those pesky "averaged group of 10" limits that my engineers seem to use as a cuddly little security blanket.

I guess I've been thinking of it as a stats problem where I've truncated one side of the population and now am trying to work backward to the "smaller" group distribution limits that would give me the right "new" range (while using that 10^.5 you mentioned to get to the wider individual range).

I should be able to make this into 2 equations with 2 unknown, but for some reason the reverse equation is eluding me.
1. initial limits + assumptions goes into CLT and out spits smaller limits
2. different smaller limits + same assumptions goes into ??? and spits out new initial limits

I should then be able to take the "new" limits, put them into equation #1 and get the smaller limits I want. If I try a goal seek or just invert the existing equation I get garbage data and I can't figure out why...
 
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