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  Statistical Techniques and 6 Sigma
  Specifications on x & r charts

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Author Topic:   Specifications on x & r charts
Bill Smith
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posted 03 December 1998 02:29 PM     Click Here to See the Profile for Bill Smith   Click Here to Email Bill Smith     Edit/Delete Message   Reply w/Quote
Per QS9000 SPC manual, are the specification limits they apply to the average chart the same as the specification on blue prints ?

If so, the B/P specifications should apply to all individual parts, not the average of 5 parts - right ?

Our software package from Quality America,SPC PCIV, does use individuals when calculating the Cpk, but allows for a moving range of chosen subgroup size to be used in place of a standard deviation. I have seen a difference of 0.5 in the Cpk values using both measures of variation when the distribution is non normal. Why is this ? Sample size was 2000.

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Roger Eastin
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posted 04 December 1998 09:24 AM     Click Here to See the Profile for Roger Eastin   Click Here to Email Roger Eastin     Edit/Delete Message   Reply w/Quote
Wait a minute...Cpk only applies to a normal (or near normal) distribution. That has been one of the main criticisms of it as a "one-size-fits-all" metric. If your distribution is non-normal then how you compute standard deviation becomes VERY important. I would suggest that you pick up a good book on SPC (by Wheeler, Montgomery or someone of similar caliber). There are other metrics you can use. If I understand your message right, do NOT associate spec limits (which apply to individual parts, as you point out) with control limits (that apply to averages). This is one reason why I would pick up a good SPC book if I were you. Spec limits should only apply to Cpk calculations (unless you are using a special type of chart) for normally-distributed data.

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Don Winton
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posted 04 December 1998 11:20 AM     Click Here to See the Profile for Don Winton   Click Here to Email Don Winton     Edit/Delete Message   Reply w/Quote
Roger,

I agree with everything stated. I do not know what the 'QS9000 SPC manual' is trying (maybe, maybe not) to imply by associating spec limits with control limits. There are alternative process capability calculations for non-normal distributions, but I do not have them handy right now. Will try to find and forward. Along with your list of books (all very good, by the way), I will add another. Bill may want to try Grant and Leavenworth's Statistical Quality Control as well.

Regards,
Don

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Don Winton
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posted 09 January 1999 08:37 PM     Click Here to See the Profile for Don Winton   Click Here to Email Don Winton     Edit/Delete Message   Reply w/Quote
As Promised:

Methods for Handling Non-Normal Data

If the data is non-normal, another method must be used in order to have a valid process capability study.
1. Transformation of the data and calculate the process capability indices using the transformed data.
This requires knowledge of the appropriate transformations. It is also time consuming because the normality assumptions must be checked after each transformation.
2. Utilize Johnson or Pearson distribution fitting techniques and determine the process capability from the appropriate percentage points of the distribution.
3. Ford Motor Company demonstrates what they call Estimated Accumulative Frequency and the Mirroring Technique to handle non-normal distributions.

Regards,
Don

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Steven Sulkin
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posted 12 January 1999 04:37 PM     Click Here to See the Profile for Steven Sulkin   Click Here to Email Steven Sulkin     Edit/Delete Message   Reply w/Quote
Bill- your question may also be pointing to stacking. Statisticians: please feel free to jump in here.

_____________________________________________

You cannot use a spec designed for individual units
to validate the average of a sample.

_____________________________________________

When you average a sample you do what is called stacking. That is, you shrink your variation by a factor of SQRT of the sample size. To get a representative spec you will have to make the same adjustment. Thus,

If your spec is for individual units and you want to adjust it for comparison against an average of five, divide the spec by the SQRT of five.

TRY THIS:
In order to see how this works, do a capability study using the individual spec and individual measures. What is your Cpk?

Now, do a capability study using the individual spec and averages of five. What is your Cpk?

You should find a big improvement when you use averages. Why? Your distorting your results by using averages (shrinking your process variation by a factor of SQRT 5).

If you make the correct adjustment to the spec you should get a comparable Cpk.

-Steve.

[This message has been edited by Steven Sulkin (edited 01-12-99).]

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Don Winton
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posted 13 January 1999 11:00 AM     Click Here to See the Profile for Don Winton   Click Here to Email Don Winton     Edit/Delete Message   Reply w/Quote
Bill,

In response to the same question at the Niles Stats forum, B. Duffy replies:

Snip:

In reply to: QS-9000 and SPC Requirements posted by Marcy White on January 12, 1998 at 11:58:08:

I flipped through Chapter II Section 1 of the QS 9000 SPC manual (1995 version) and did not see anything pertaining to spec limits on the X-bar chart. Do you mean control limits? (Spec limits should not be drawn on a control chart--but they come into play in capability analysis.)

If you mean control limits, then the answer is no, the limits can not be compared to the specs on the drawing. As you state, spec limits apply to individual measurements. If you are plotting an X-bar chart, you can "convert" the standard deviation of the chart to an individuals basis by taking the distance between the X-bar limits, dividing by six, and then multiplying by the square root of the sample size. I think you can also take the value of R-bar (central line of the range chart) and divide it by the value of d2 that corresponds to the sample size.

I believe this estimate of the process standard deviation could be used in the Cpk calculation-you need not generate an individuals chart. If you do create an X-mR chart, I'd suggest a moving range size of n = 2, no bigger.

If the process does not display statistical control, the two methods may not yield similar results, so your Cpk will vary. (You may want to compare methods using data known to be random, say from a random number table, to get a feel for the difference.)

Finally, the functional form of the process distribution is not important--unless you're determined to estimate percentages of nonconforming product as tail areas of the distribution beyond the spec limits (not recommended). What matters is whether the process is stable. A stable process is reproducible within limits. Thus, the Cpk can be used as a prediction (very important to customers). In this case, the control chart, if it shows stability, fulfills requirements laid down in the theory of knowledge.

mbd

End Snip

I think B. Duffy makes some points well.

Steven,

quote:
When you average a sample you do what is called stacking. That is, you shrink your variation by a factor of SQRT of the sample size. To get a representative spec you will have to make the same adjustment. Thus, If your spec is for individual units and you want to adjust it for comparison against an average of five, divide the spec by the SQRT of five.

Your response and following example are correct, although I have not heard it referred to as stacking. I would, however, recommend adjusting the standard deviation by the SQRT of the sample size, not the spec. Perhaps I misunderstood.

I would add that when doing a process capability study, careful consideration must be paid to sigma. There are corrections available for individual measurement sigma when the sample size is less than 100. See Grant and Leavenworth Statistical Quality Control, 6th edition, pp. 107-108 and Table C in Appendix 3, same source.

As far as normality testing goes, I also found this at Niles:

Snip:

Jack Tomsky Replies:

Probably the best normality test, in the sense of detecting non-normality, is the Wilk-Shapiro test. Theoretically, this test involves the ratio of two estimates of the variance of a normal distribution. The first estimate is the square of the minimum variance linear unbiased estimate of the standard deviation of a normal distribution, based on a linear combination of order statistics. The second estimate is the usual sum-of-squares estimate. Since these involve coefficients that need to be obtained from tables, the easiest way to perform this test is through a statistics software package. "Distance" tests such as Kolmogorov-Smirnov and chi-square have been shown to be generally inferior to Wilk-Shapiro.

End Snip

Regards,
Don

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