# Cp & Cpk about a diameter measured with min.,max. and average values.

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#### Mark_Navigator

Hi all,

I've a question about the process performance related to a dimension with form error, like a diameter with minimum, maximum and then average values got.
The minimum and maximum values, have to be aspected to be conformed to smallest extreme and maximum extreme value modeling?
It's not possible to have a normal distribution with min. or max. values? (never found till now)
How can be assumed the analysis if the average values give a normal distribution with very good performance indexes, and extreme values are closer to the limit, and specially one of them are out of specifications?
Simply describing to designer that the average diameter is ok, and the single extreme values not?
So does he have to evaluate if also extreme values are significant for the requirements?
Is this analysis right?
Here as attachment the measures and Mini Tab analysis (all distributions result non normal, due to not sensitive measurement method, but the average distribution can be assumed normal, instead of extreme values assumed as mentioned above).
Thank you for everyone can help me to solve this doubt and gives me higher knowledge.

#### Attachments

• min.max.average steinless steel ring diameter real case.xlsx
38.4 KB · Views: 267

#### bobdoering

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You have a couple obvious problems. First, your tolerance is 100 microns, and your measurement system is only 10 microns. If you did a gage R&R I bet your system has way too small ndc. Secondly, your within-part variation (roundness) is using up half of your tolerance. (See column of data I added to the attached chart.) Very, very difficult - but not impossible to control. In order to control it, however, you would have to use X hi/lo-R methodology. Further, most of your variation in your charts is measurement error. Tool wear is the continuous uniform distribution. You need some time-ordered data in that chart form to figure out what your process is doing. Once properly controlled with that chart, you capability is easy: (USL-LSL)/(UCL-LCL) where (UCL-LCL) is generally 75% of the tolerance. That ONLY works when properly controlled, however.

Remember, average doesn't count. Your entire part needs to be in spec. Rule of life: If you measure one place, you will measure a good one...and your customer will measure a bad one. And the average won't tell you either.

#### Attachments

• min.max.average steinless steel ring diameter real case.xlsx
41.7 KB · Views: 225
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#### Mark_Navigator

About the measurement system and tolerance the issue is that are both with 10microns of resolution, and theorically the measurement should have a resolution 1 degree lower than the requirement, but the handling of a simple caliper is easy and cheap, why the further measurements should be affected by many errors in this way?
The roundness is not diameter max.-min., the definition is different and referred to the center of the circle, so getting it with the diameters is approssimative and have to be considered the half value! (Dmax-Dmin/2)
Assuming to consider only all dimension and not the average as a rule, is not right in my opinion and experience made, because for some functional and assembly reasons the average can simply be accepted and required (e.g. an oval of thin steinless steel ring after the assembly can became perfectly round), so could be only necessary to have in the drawing a note like "average".
But finally, the analysis that I published, canb e assumed correct to give the whole situation about the dimension? (so with min.,max. and average analysis?)
It's normal to get non-normal distribution for min. and/or max. values of a dimension, respect the average values with normal distribution?

#### bobdoering

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About the measurement system and tolerance the issue is that are both with 10microns of resolution, and theorically the measurement should have a resolution 1 degree lower than the requirement, but the handling of a simple caliper is easy and cheap, why the further measurements should be affected by many errors in this way?

10:1 is rule of thumb, ndc is more accurate way to determine statistical resolution needed. Cost and ease are not the issue - resolution is. Without adequate resolution, your data will be "chunky" making our distribution le clear.

The roundness is not diameter max.-min., the definition is different and referred to the center of the circle, so getting it with the diameters is approssimative and have to be considered the half value! (Dmax-Dmin/2)

Yes, in GD&T, roundness is radius error, but roundness is still what is affecting your measurement error. But - in your Ppk calculation it is affecting it as a diametric error.

Assuming to consider only all dimension and not the average as a rule, is not right in my opinion and experience made, because for some functional and assembly reasons the average can simply be accepted and required (e.g. an oval of thin steinless steel ring after the assembly can became perfectly round), so could be only necessary to have in the drawing a note like "average".

If fit is the issue, then average has nothing to do with it - only MMC. If "passing the spec" is the issue - which it truly the case (that is why the spec is part of the calculation) then min and max are both critical issues. If average is a drawing note, then it becomes the spec and justifies its use.

But finally, the analysis that I published, can be assumed correct to give the whole situation about the dimension? (so with min.,max. and average analysis?)

It can - much better than the wrong, traditional way of taking one diameter per part and plotting the results. My point was to illustrate why your Ppk was so bad - you are using up 50% of your tolerance with "with-in part" variation - leaving little for between part variation. I also described a better way to track and control the problem as it is manufactured.

It's normal to get non-normal distribution for min. and/or max. values of a dimension, respect the average values with normal distribution?

Machining - specifically tool wear - is non-normal to begin with. Min-max looks non-normal because it is describing multimodal error. However, it does describe the full error.

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#### Mark_Navigator

Machining - specifically tool wear - is non-normal to begin with.

What do you mean with this phrase? (sorry, probably the syntax is not easy to understand for me, an italian)
By the way, thank you for all the explanations.
Now I've understood what you mean with "with-in part" tolerance usage, and it's simply right what you said, and it'll be very useful for me to make designers to understand better why became so difficult to be in specifications with this kind of situation.

#### bobdoering

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You may want to refer to my blog for more details. Bottom line is tool wear is a time function, where the dimension increases (OD) or decreases (ID) at a rate over time. At some point an adjustment is made in the opposite direction of the wear, and the wear begins again. This wear-adjust-wear-adjust cycle generates a sawtooth curve, and its distribution is a continuous uniform distribution. Non-normal, non-random, and dependent variation.

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#### zancky

my modest opinion

according to ISO 14405 (when will we see on the drawing?)
You should find on the drawing one of the following mark requirements:

(GG)
for each dimension the average has to calculated with minimum square error (i.e. a lot of data for each part not (max+min)/2)

(E)
the dimension must be within the tolerances on every point, so the min and the max.
As Cpk is an indicator about the failure rate the max and the min value has to be considered (Cpk on both)

(SD) the (max+min)/2 has to considered. that's what You did

etc

previously the use of the ISO 7167 or the 8015 told you if you were in the case (GG) or in the case (E).
For the US market the taylor principle must be applied (i.e. (E))

#### bobdoering

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Interesting points.

In my opinion, Min/Max taken over at least 8 data points per circular section (slice) is a clear representation of the part dimension. Very similar representation to - but clearer to visualize than - GD&T Diameter and circularity together. That should accommodate most within-part form errors (lobing being the most significant for machined parts - not so much for stamped parts - that is why considering the process is critical). Now, to be even more accurate and descriptive, you should have 3 slices per cylindrical section to include the taper error, too.

Not sure how the ISO committees came up with their notions.

By the way, X hi/lo-R charting methodology can also control taper. Try that with a Shewhart chart!

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#### Mark_Navigator

Thank you Bob for the explanations.
However (to all people), as told to me by an italian statistical engineer regarding min. and max. values of a dimension, those conditions should be analysed by using min. and. max. extreme values distribution models, and expected that values assume these distribution characteristics, isn't it?

Another issue: and about Natural tolerance of a process with min. max. and average values of a dimension, how have to be assumed and calculated? Or simply it have to be given for every type of dimension?

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#### Bev D

##### Heretical Statistician
Super Moderator
As always physics precedes and supercedes ALL distributional models. The data should tell us how to analyze it. plot your data (raw data; actual results) and then think about it.

Distributional models are man made constructs.

The real strength of any analysis comes from understanding what the process under study is actually doing...