How to derive Average Range Factors, A2, D4, D3? Statistical Process Control
Michael Furlan
have yet seen) has the table of values for the factors for computing
control chart limits, but none show how those values were derived.
I don't expect anyone here to explain this to me, but I would
appreciate being pointed to a book, ideally still in print, that
explains the theory behind that table.
Craig Cochran
I believe those factors come directly from the man himself, Walter
Shewhart. He's the guy that came up with the notion of the Central
Limit Theorem and control charting. The factors come very close to
approximately the real calculations, if you run a comparison. See if
you can round up a copy of this:
Shewhart, W.A., "Economic Control of Quality of Manufactured Product,"
D. Van Nostrand Co., New York, 1931.
It might not be in print, but a large university library would
probably have a copy.
Good luck,
CC
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Craig Cochran
Center for International Standards & Quality
Georgia Institute of Technology
www.CISQ.gatech.edu
Duane Allen
> Michael,
>
> I believe those factors come directly from the man himself, Walter
> Shewhart. He's the guy that came up with the notion of the Central
> Limit Theorem and control charting. The factors come very close to
> approximately the real calculations, if you run a comparison. See if
> you can round up a copy of this:
>
> Shewhart, W.A., "Economic Control of Quality of Manufactured Product,"
> D. Van Nostrand Co., New York, 1931.
>
> It might not be in print, but a large university library would
> probably have a copy.
>
A commemorative edition of Shewhart's book is available from ASQ. I
found the book less informative than many modern texts on statistical
process control. In the book, Shewhart is providing a rationale for
using statistical methods and charts to control industrial processes.
I would not recommend it for the average student, except for a brief
examination. And it is not the source of the estimators for control
control charts as you speculate.
The d2 factor dates back to the early 1900's. It is in a statistical
journal. The article relates the range to the standard deviation. The
article is interesting in the integral values were calculated, I
presume, by hand.
Duane Allen
Duane Allen
Some of the factors are based on were the +/- 3 sigma limits would exist
for a student-t distribution for a given sample size.
According to Tom Vaden, there are actually several versions of control
charts tables in existance. Some have errors. If you're a graduate
student, there is potential paper.
Apparently, also there are tables that are based on +/- 3 sigma, which
is about 99.73% of the area under an normal curve, and tables that are
based on a 99.90% of the area.
Duane Allen
Robert Gerst
available from Amazon, SPC Press and ASQ.
The formulas for A2 etc are shown using the d2 bias correction factor.
"Michael Furlan" <mi...@thecivilwargroup.com> wrote in message
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John Duffus
I don't know if this will get you any further forward but I came across the
following at
http://www.stat.jyu.fi/saskurssi/sashtml/qc/chapc/sect9.htm
d2 = the integral from minus infinity to infinity of the following function:
1 - (1-P(z))^n - P(z)^n with respect to z
where P(z) is the cumulative standardized normal distribution and n is the
sample size.
I did a numerical integration in Excel over 200 points using NORMSDIST and
the result agrees with the tables within 0.5%.
Unfortunately this doesn't give me any more insight into the rational.
You might try one of the statistics groups. d2 is the expected value of
the sample range of n normally distributed variables with SD 1, and would
have wider statistical application than just control charts.
John Duffus
"Michael Furlan" <mi...@thecivilwargroup.com> wrote in message
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nospam@all
Shewhart was a statistician, but I bet he had help developing the factor tables.
I found the links below which I thought was interesting...
http://www.itl.nist.gov/div898/handbook/pmc/section1/pmc11.htm
http://www.sytsma.com/tqmtools/ctlchtprinciples.html
http://www.snqc.org/INFORMATION/QualityLeaders.htm#Walter%20A.%20Shewhart
http://lucent.netlabs.net/work/family/docs/shewhart.html
http://www.newprometheans.org.uk/aimp.htm
under the heading ...
THE IMPORTANCE OF THE TELEPHONE
Jim Winings
http://www.sixsigmaspc.com
John Duffus
formula, what is the reasoning behind the formula? It's just curiosity
really -it has no practical significance in using control charts. It has
been suggested that Shewart gives the derivation in "Economic Control of
Quality of Manufactured Product", but not having easy access to a good
library I cannot verify this.
John Duffus
"Robert Gerst" <rober...@home.com> wrote in message
news:X7qha.586878$Yo4.42...@news1.calgary.shaw.ca...
Michael Furlan
<john....@telus.net> wrote:
>How d2 is derived is the main question.
"There is a well known relationship between the range of a sample from
a normal distribution and the standard deviation of that distribution.
The random variable W=R/sigma is called the relative range. The
parameters of the distribution of W are a function of the sample size
n. The mean of W is d2. Consequently, an estimator of sigma is R/d2.
. . .
A2=3/d2 * square root of n"
p. 174-175 "Statistical Quality Control", Montgomery.
How is the relationship "well known" when nobody here seems to know
it? <g>
Duane Allen
> How d2 is derived is the main question. ...
The d2 factor is derived from W, which dates back to an early 1900's
statistical journal. The article relates the range to the standard
deviation. I haven't seen my copy of the article for several years or
else I would cite the reference.
Duane Allen
Michael Furlan
<comput...@sprintmail.com> wrote:
>The d2 factor is derived from W, which dates back to an early 1900's
>statistical journal.
Found elsewhere:
* Subject: Re: Relation of R-bar/d2 to standard deviation
* From: Alson C H Look <al...@pacbell.net>
* Date: Mon, 06 Jan 2003 22:46:14 -0800
Hi,
The derivation of d2 is rather complicated to put in a post. You can
read its derivation in Irving Burr's book Statistical Quality Control
Methods.
It is interesting to note that the upper bound for d2 according to
Kendall and Stuart's Advanced Theory of Statistics is 1.155 (for any
distribution) for samples of size two. For the normal distribution it
is 1.128. The original source for d2 is Tippett's 1925 paper: On the
Extreme Individuals and the Range of Samples from a Normal Population.
Biometrika vol 17 1925. Tippett calls d2 w.
The derivation requires according to Burr approximate triple
integration of what are called order statistics.
A very concise derivation of control chart constants can be found in
Tom Ryan's book Statistical Methods for Quality Improvement.
Alson
al...@pacbell.net
From:
http://deming.ces.clemson.edu/pub/den/archive/2003.01/msg00045.html
Terry Harris
"Where DoThe Control Chart Factors come From?", Dr. Edward Schilling, 56th
Annual Quality Conference Transactions.
Dr. Schilling can be reached at egs...@RIT.edu. Maybe he could send you a
copy.
"nospam@all" <jim_...@yahoo.com> wrote in message
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