Clarifying Control Chart Limits

[John <jsdwd@ksc.th.com>]

on 1/2/03 9:08, Kromkowski@aol.com at Kromkowski@aol.com wrote:
> In the end, here is the point I tried to make 6 years ago:
> 
> When we don't know really the origins of the X/Mr chart, when we are really
> fluffy about the theoretical basis,
> when most of the DEN and even some acknowledged experts
> (like Wheeler, let's name names) are kind of quesy (or so
> is the appearance)

As I said in my post of a week or so ago, the sub-group formation is the key
to setting limits. Grant and Leavenworth cover this topic well in their book
and Wheeler does also.
 
Some basics:

1. One wishes to provide a basis for reacting to a given process output
should that make economic sense.  It does not make economic sense if a given
output is merely another of the endless stream of expected system outputs.
In that case the economically sensible thing to do is to alter the system.

Reacting to a given output does make economic sense, however, if that point
is unusual (i.e. special, outlying, anomolous, etc....put in your favorite
word here).  The cause of the specialness, anomoly or whatever can be
(presumably) identified and isolated.  That is, that particular occurrence
point is not '...just more of the same.'

2.  The Statistical Control Chart was a tool devised by Shewhart  to provide
the basis for making the above distinction.  As such it is meant to identify
special causes (let's use that term henceforth).  The errors that can be
made are to 1.) call something special when it is (economically) 'more of
the same.' or to 2.) treat the process as an uninterrupted stream when
something unusual (special) happened.

Shewhart's writing spells out his thought process in unraveling this problem
of making an appropriate distinction.  He devised control limits based on
three standard deviations.  While they are reliant on some theoretical
underpinnings, they are empirically based.  Wheeler points this out with his
'empirical rule' and Shewhart points it out himself in Statistical Method
from the Viewpoint of Quality Control.

3.  We wish then, a set of limits that will encompass only the common cause
vairation (but all of it, lest we make mistake #1).  How shall we go about
getting such limits?  This is where sub-grouping comes into play.

It is likely that special causes are more apt to be spread out over time and
across manufacturing conditions (i.e. Different shifts, different machines,
etc.)  (Be aware that this is not always true!)

Thus estimates of variability that encompass long periods of time or a
variety of manufacturing conditions are apt to include both common cause and
special causes sources of variation.  We wish to avoid that in setting
limits.  So usually a sampling (or subgrouping) strategy is formulated that
attempts to encompass only common cause variation within the subgroup and
special causes will thus be seen (if they be seen at all) between subgroups.
This is the usual basis for so-called 'rational sub-grouping'.

Subgroup sizes are kept small (usually 4 or 5) so that they have less chance
of encompassing a special cause.  Also, if the range is used to estimate the
subgroup variation, it loses power as subgroup size increases (for obvious
reasons).  The sub-group standard deviation can be used, but subgroup sizes
should still be kept rather small.

Of course, throughout all this, process knowledge is vital.  The behavior of
the process and the economics of sampling are key considerations and a
process expert with the statistical knowledge of what is being attempted is
the best person to figure out how best to subgroup.  It is usually not that
simple and does not lend itself to formulaic answers, 'expert' systems, and
the like.  They've been tried.

4.  In an X-bar chart, this sub-grouping is more straightforward than with
the XmR chart which is a little more brazen in its assumptions.  It is used
when data cannot be conveniently sub-grouped.  A kind of 'quasi-subgroup' is
formed by using successive ranges and that is used to set the limits.  The
thinking behind it is really the same though.

The root mean square variation of the data will includes all variation
including that due to special causes.  In the presence of special causes,
limits will therefore be wider than if they were based on common cause
variation alone.  That makes the chart less sensitive to special causes.  An
estimate based on the moving range will thus be more useful.  If there are
no detectable special causes the two  estimates should be roughly the same.

With regard to d2, sorry to disappoint, but it's not mysterious at all.  It
is covered in Acheson Duncan's book, "Industrial Statistics" and, as has
been pointed out, in other books as well.  "d2" is a correction factor for
estimating the standard deviation from the range.  It is derived from a
sampling distribution of "d*",s that are generated from estimating a given
standard deviation (s=1) from subgroup ranges from subgroups of varying
sample sizes.  The d2 values are the averages for a given sample size from
those sampling distributions.


Best wishes to all for the New Year,

John Dowd
jsdwd@ispwest.com