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  Statistical Techniques and 6 Sigma
  Calculating Limits- 3S or 2.66xR-Bar?

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Author Topic:   Calculating Limits- 3S or 2.66xR-Bar?
Steven Sulkin
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From:Columbus, Ohio
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posted 02 September 1998 05:39 PM     Click Here to See the Profile for Steven Sulkin   Click Here to Email Steven Sulkin     Edit/Delete Message   Reply w/Quote
A question has been posted to me. What calculation should we use to calculate control limits and what makes them different. We have seen calculation that use a weight factors (eg. 2.66 for a moving range), or using the 3 times the standard deviation. They generate very different results. Explain why?

Thanks in advance.

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Don Winton
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posted 21 September 1998 05:26 PM     Click Here to See the Profile for Don Winton   Click Here to Email Don Winton     Edit/Delete Message   Reply w/Quote
Steven,

I will try to shed some light on this for you (maybe?). The different weight factors are determined by the sample and method used to estimate standard deviation. It is important to remember that control limits are based on an estimate of the standard deviation. The actual standard deviation of a population is not known unless the entire population is known and measured, which it is not in most cases where SPC is used. If sigma is estimated from the Rbar values for subgroup size 4 it is very different if sigma is estimated from sbar for the same subgroup size.

Example equations for estimating sigma from Rbar, sbar, sigmabar_rms and sigma_subR:

d2=Rbar/sigma
d3= sigma_subR/sigma
c2= sigmabar_rms/sigma
c4= sbar/sigma

The actual derivations for the statistics involved are too detailed to go into here. I would suggest Process Quality Control by Ott and Schilling, Statistical Quality Control by Grant and Leavenworth or JuranÔs Quality Control handbook if you are interested.

The 2.66 factor for moving range is derived from Rbar (average of the moving range) and a subgroup size of 2 (Measurement 1 ÷ Measurement 2, Measurement 2 ÷ Measurement 3, etc).

Also, remember this: U/LCL = Xbar +/- (3*sigma) is not the same as using the weighting factors and will yield erroneous results. Try the PDF zone at this site or www.qualitymag.com for additional reference material.

Does this help or did I just muddy the water?

Best Regards,
Don

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Steven Sulkin
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From:Columbus, Ohio
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posted 22 September 1998 08:09 AM     Click Here to See the Profile for Steven Sulkin   Click Here to Email Steven Sulkin     Edit/Delete Message   Reply w/Quote
Don writes....
"Also, remember this: U/LCL = Xbar +/- (3*sigma) is not the same as using the weighting factors and will yield erroneous results."

This is exactly my question. Why is this calculation erroneous? Whether we use 3sigma or weight factors, we are still estimating the spread of a process, so why do we not get the same result?

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Don Winton
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From:Tullahoma, TN
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posted 22 September 1998 11:52 AM     Click Here to See the Profile for Don Winton   Click Here to Email Don Winton     Edit/Delete Message   Reply w/Quote
Steven,

Perhaps I was too general in my statement. As stated, "erroneous results" is not exactly accurate. More correctly stated, "using U/LCL = Xbar +/- (3*sigma) can lead to an incorrect decision as to whether a process is in or out of control." More often than not, using (3*sigma) yields wider control limits. I did not mean to imply that (3*sigma) cannot be used, only to use it with caution (Besides, it is sometimes easier). I was also incorrect when I stated that "If sigma is estimated from the Rbar values for subgroup size 4 it is very different than if sigma is estimated from sbar for the same subgroup size." It should have stated "When sigma is estimated from the Rbar values for subgroup size 4 the method is different than if sigma is estimated from sbar for the same subgroup size." I apologize for the misunderstanding. I must learn to proofread these things before posting. Anyway, back to the issue. From your first post, I am assuming that you are using an X-mR or mX-mR method to monitor your process. This may not be correct, but if so, refer to www.qualitymag.com/articles/jan98/0198wh.html for an article on this topic.

As for the use of weight factors. In order to draw conclusions about an unknown population, it is necessary to rely on samples. These numerical values, Xbar and s (or whatever), summarize the information contained in the sample data. They are referred to as a statistic of the sample and may be used to estimate the corresponding parameter of the unknown population, in this case mean and sigma. The parameters of a population are not known unless the data of the entire population are known. When the standard deviation is calculated from the X values, this is a statistic (s), not a parameter (sigma). That is why you do not get the same results. This method is demonstrated in Shewhart's normal bowl experiments in "Economic Control of Quality of Manufactured Product." Shewhart's normal bowl contained a population of known parameters. By sampling from this bowl, the weight factors could be derived. Of course, this is a very simplistic explanation of the subject and is discussed in detail in Grant and Leavenworth's book (see above). Try a local library to obtain a copy of any of those above for the details of weight factors and their use.

So from all the above, it should be more technically correct to state "using U/LCL = Xbar +/- (3*s) can lead to an incorrect decision as to whether a process is in or out of control." Don't you agree? :>)

Best Regards,
Don

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