Control Limits for Individuals Charts

["John McConnell" <wysowl@msn.com.au>]

The discussion on whether to use Root Mean Square Deviation or another
estimate based on the Moving Range chart is interesting.  It gave me a sense
of déjà vu.

Many years ago when I was struggling with this subject I conducted some
experiments to determine which approach seemed superior.  The results of
these experiments seemed pertinent to the discussion.  The outcome was a
decision that the Moving Range Chart was superior because it offered more
information and, if understood and used properly, was less prone to errors
in interpretation.  Not perfect, better.

The first thing I did was to determine the aim, which was to find the better
method of separating signals from the noise.

If we look at the formulae for ave and range as well as single point and
moving range control charts, we soon notice that all the control limits for
all charts are a function of R bar.  That is, if R bar is a reliable
estimate, so too are the control limits.  To the extent that R bar is
corrupted (by special causes etc), so too will be the control limits.

A bunch of Red Beads data was broken into sets of about 50-75 data.  Limits
were calculated using both RMS Deviation and the ave of the Moving Range
chart.  First conclusion:  if the data are stable, and Red Beads data
generally are, both methods gave very similar results, which one might
expect.  But initially the control chart is in no small way a test for
stability.  So it was necessary to mess up the data a little to see how both
methods handled unstable data.

Using a playing die to randomise matters, I added some special causes to the
data and created some shifts in the process average.

The limits from the Moving Range chart were less corrupted by the presence
of special causes in the calculations.  A look at the formula for RMS
Deviation soon explains why.  Shewhart's writings (already mentioned in
these discussions) also give some pretty good clues.  It matters not one
whit how unstable the data are, if they are reasonably continuous, limits
based on plus or minus 3 RMS Deviations will nearly always result in most,
if not all the points falling within these limits.  + or - 3 RMS Deviation
is a poorer test for stability because it is less likely to separate signals
from noise.

Moreover, although the Moving Range chart also was corrupted by the presence
of special causes, many of them were obvious because they were beyond the
UCL.  Also, a single special cause in the data always resulted in two
consecutively high (generally out of limits) points in the moving Range
chart.  When the process average stepped up or down, a single spike in the
Moving Range chart was observed in most cases.

These patterns were consistent and repeatable.  In a Moving Range chart,
they must be (unless the special causes are small compared with the random
noise), because of the way we calculate the Moving Range.  Also, because of
the repeatable, consistent patterns, in most cases they were reasonably easy
to detect, even before limits were calculated.  When these points were
removed from the calculations for R bar, the limits became pretty much the
same as when the data were stable.  Again, if R bar is reliable, so too are
the limits.

The methods were then shifted to data from the workplace.  Manufacturing
data from a meat processor were used initially.  I chose data from two
processes that had been in a state of chaos, but then stabilised.  Data from
the period of chaos were used first.  When + or - 3 RMS Deviations was used
to calculate limits, all the data except for a few gross special caused fell
within limits.  When the Moving Range chart was used, both special causes
and shifts in the process ave were much more easily identified, sometimes by
points falling beyond the limits, sometimes by the repeatable patterns
previously mentioned.

The methods were then shifted to service industry data, in this case, a
bank.  As one might expect, this data tended towards stability to a greater
extent than did manufacturing data.  There were fewer signals in the data.
Still, the Moving Range chart did a superior job at identifying the
disturbances.

More data from a metallurgical plant demonstrated that the Moving Range
chart was superior at detecting over-control.  These signals were lost when
using + or - 3 RMS Deviations.  Interestingly, regular over-control often
shows up in an Individual point (and Moving Range) chart as "hugging the
centre line", the exact opposite of what we se in an ave and range chart.

My conclusion was, and remains, that the Moving range chart is generally
superior, although there are situations where the differences are minimal.
RMS Deviation measures the variation in the data, or the tendency of the
data to disperse around the mean.  Partly because it loses the information
contained in the time order sequence of the data, it is particularly poor
when there are shifts in the process average and when special causes are
regularly present.  On the other hand, a Moving Range chart is only as good
as is the estimate for R bar.  Nonetheless, my experiments demonstrated that
the Moving Range is superior at separating the signals from the noise.  If
another aim is chosen, another method might be superior.

Remember the 15th Point!

Cheerio!

John McConnell
wysowl@msn.com.au