Re: X and Moving Range

["John Constantine" <thesfg1@cox.net>]

Eileen Beachell notes:

"Dr. Wheeler did not invent the individual and the moving range chart. So,
what was the 
basis for the material in his book on the topic? What were the papers or 
textbooks that did develop the individual chart? Dr. Wheeler doesn't seem to

know or at least did not refer to them in his book."

May I take a crack at this topic...

As Dr. Wheeler notes in his writing, in seeking knowledge about the
variation in a process, we are seeking the location or mass of the data as
well as its dispersion characteristics. Why he chose not to identify the
primary source documents for the "original" XmR chart I haven't the
slightest idea. But as he refers so often to Shewhart and Deming on the
subject matter, I think that he simply have thought that it wasn't necessary
or made a choice to let that be a learning process for others who might be
interested. (By the way, I haven't asked him.)

He does go into the subject of the origin of constants in great detail on p.
237 of Understanding Statistical Process Control pointing out that the
constants are linked. Always looking for both location and dispersion, he
states that the XmR chart "...is frequently the most sensitive chart for
this (periodically collected) type of data".

I have also found a wealth of discussion on these matters contained in Dr.
Deming's book SOME THEORY OF SAMPLING, original copyright 1950, published by
John Wiley & Sons, Inc., and later published by Dover by Dover Publications,
Inc. in unabridged and unaltered form in 1966.

I would point the reader to Part V - for an in-depth discussion of theory
for design and analysis in general, and to Part V Chapters 15 and 16
specifically for detailed consideration of the "DISTRIBUTION OF THE VARIANCE
IN SAMPLES FROM A NORMAL UNIVERSE" and "TESTS FOR HYPOTHESES IN NORMAL
THEORY".

(I have always wondered why he (Deming) supposedly said that about half the
sample would be above the mean, and half below. I believe this chapter gives
the answer in relation to the application of theory vs. the actual data
themselves. It turns out that in 51 of 100 samples, the mean is contained
therein. Perhaps this is why Deming used the term "about half".)

In this section, there is Fig. 84; originally published by Deming and Birge
in 1934; Deming attributes the idea to a figure in Shewhart's Economic
Control of Quality from 1931. The figure serves to illustrate the problem of
small samples. Deming also states that Shewhart (p. 52 of Statistical Method
from the Viewpoint of Quality Control - 1939) "...demonstrates how futile it
is to attempt to estimate the magnitude of sigma from samples of 4, but that
samples of 100 are excellent, while samples of 1000 give practically perfect
results.", p.564. He goes on to display and discuss the computation of the
control limits or error band for samples from a known universe, wherein he
sets out in a table the mean, standard deviation, xbar, s, s-squared and
range parameters for each sample size n.

>From p. 565, the constants which we may be familiar with, namely c2, d2,
etc., again referencing Shewhart. He also lays out the theoretical
relationships in terms of "expected values" of s and R., and while he does
not calculate the d2 constant, he does give the calculation of the c2
constant . Why is this important? It is used later in the construction of
control limits for control charts.

Because, as he points out "it is useful to adopt and test the hypothesis
that the quality is varying like numbers being drawn from a bowl in the
ideal-bowl experiment". Randomness, in other words. If they behave in such a
manner, the process is said to be In statistical control . But, ...

("...occasionally at least, the level (mean) of quality slips away to one
side or the other...and slips far enough to cause concern." The hypothesis
H2 (out of control w/assignable causes) is also similar with reference to
dispersion (sigma)."

Deming follows this with tests for the various hypotheses, using control
charts. "There will be a chart for xbar and one for either R or s (but not
both)." 

>From p. 566, "The theory of the 3-sigma limits is as follows: i. the mean is
of samples of n are distributed with standard deviation sigma divided by the
square root of n.; ii. practically all of any ordinary distribution is
included within the compass +/- 3 standard deviations, and any point beyond
the 3 sigma is taken as an indication of an assignable cause. For the xbar
chart, the position of the s-sigma control limits will be xbar +/- 3
sigma/sqrt n"

sigma is estimated from the average value of R or the average value of s.
Thus,  sigma = Rbar/d2 if the R-chart is used, sbar/c2 if the s-chart is
used. The 3-sigma control limits for the xbar chart are thus computed as
xbar+/- 3/d2sqrt n times Rbar if the R-chart is used". (Or substitute c2 for
d2 if the s-chart is used.)

Finally, from p. 567, Deming points out that the xbar chart is used to
discover assignable causes that affect the level of xbar, while the R-or
s-chart is used to discover assignable causes that affect the dispersion of
quality.

I hope this helps and does not hinder anyone from seeking further knowledge
on the subject and the source documentation behind the relationship of the
Range and the Mean. My apologies to Jim for the inability to provide
scientific notation for all the formulae contained above.

________________
John Constantine
thesfg1@cox.net
Phoenix, AZ