Tom Ryan's reply to the SPC theory point

[tpr42345@aol.com]

I want to clarify my position on the use of an X-chart in place of a c-chart. 

The main problem in making such a substitution is the following.  As you 
know, a
c-chart is an attribute chart and an X-chart is a chart for measurement data. 
 
When we use the latter we naturally use 3-sigma limits because we assume that
we have a symmetric distribution, and ideally the distribution should be 
close to a normal distribution. If not, the chart will have properties much 
different from what is assumed.

We should generally not use 3-sigma limits for attribute data because
the appropriate distributions are skewed, and may be greatly skewed.
When we start applying symmetric limits to Poisson (or nearly
Poisson) data, we can have all kinds of problems, including the 
nonexistence of a lower control limit (LCL). This is a serious problem
because we use attribute charts primarily for the purpose of
detecting improvement -- as reflected by points falling below the
LCL -- but obviously that can't happen if we don't even have a LCL!

If we are going to substitute an X-chart for a c-chart, which indeed might
be a good idea if the Poisson assumption is not met, we are immediately
confronted with the problem of how to determine the control limits.  This
perhaps unanticipated obstacle cannot be avoided simply by using
3-sigma limits.  Although the Poisson model might not be appropriate,
we can't act as if we have a symmetric distribution.

Simplicity is fine as long as the results are reasonable, but we cannot
assume that almost everything is normally distributed, or to act as if
non-normality doesn't matter.  We don't teach statistics that way, and 
statistical methods should not be used in such a manner.

I know that some of you on this board are indeed rather analytical
and are inclined to perform simulations to obtain insight.  If you doubt
what I am saying, investigate it for yourself.

Also, there are some results in an unpublished manuscript (not written
by me) that support my position regarding the substitution of an X-chart
for a c-chart.  If you are interested, drop me an e-mail and I will give 
you some of the technical results.


Tom Ryan