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I have read several publications where estimations of standard deviation are calculated using factors such as d2, a2, etc. For example, Wheeler, in his book, Understanding Variation, uses RBar x 3.27 for calculating the upper control limit for a range chart and the value (RBar x 2.66) added or subtracted from XBar to calculate upper and lower limits for an X chart.
Where do these factors come from? Is there another method I should use to calculate limits?
The special statistics based on the range were developed over 50 years ago.
From pp.145 Quality Control and Industrial Statistics (Duncan 4th edition) is the following:
“In 1950 P.B. Patnaik showed that the square of the average range has a distribution that is approximately of the form of a Chi Square distribution. For the case in which the average range is the average of ranges of g random samples from normal universes with a common variance, each sample containing m cases, Patnaik worked out conversion factors d2 and degrees of freedom for various values of g and m.”
His original paper is “The Use of Mean Range as an Estimator of Variance in Statistical Tests” – Biometrika Vol 37 (1950) pp. 78-87
There has been a great deal done with this since his original paper and I suspect that if you check either the internet or statistical references and look under topics such as “variance estimator” and/or “mean range use” you will probably find one or more technical historical treatments of the subject that will guide you through the finer points of the derivation of d2.
Click here to read a thread on the same subject:
It is not in that thread, but I remember after it that someone posted that, in fact, those constants are mathematically derived and pointed an article where it was explined. You can try a search.
Anyway, I think you should stick with that method (with the factors) to calculate control limits.
In addition to the other responses already given, the formulas for the control limit factors can also be found in the ASTM (American Society for Testing and Materials) Manual on the Presentation of Data and Control Chart Analysis. I got my copy from ASQ’s Quality Press. Supplement A, beginning on page 136 of this manual, presents all the mathematical relations and formulas used to derive the various control limit factors.
In addition, this manual also shows the correct methods of calculating control limits.
Hope this helps.
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