xBar-R versus X-chart control charts

[Steven_S_Prevette@RL.gov]

Dan, in his recent email, questioned my use of the standard deviation
calculation for control charts rather than using xBar-r (which Dr. Wheeler
definitely recommends).

There are specific reasons I have chosen to use the x-chart (with sigma as
the basis for the control limits) rather than xbar-R.  The advantages of the
calculated standard deviation are:

1.  In a sample of a stable process, the sigma formula gives a more accurate
value for the dispersion of the data than does the average range.  Dr.
Shewhart does document on page 287 of Economic Control of Quality of
Manufactured Product that the sigma formula is the most "efficient" estimate
of standard deviation.

2.  The sigma formula is much easier to use with PC's and spreadsheets than
the average range.  This is the opposite of the situation even 20 years ago,
where the average range was easier to calculate with mechanical adding
machines and slide rules.

3.  Early attempts to use the x-bar R charts (with subsample size of 4) at
Hanford led to great confusion from management.  Trying to explain what the
Range is, and that averages are plotted rather than individual values led to
major problems.  Also, management here insists on grouping data by calendar
month, not by how ever many days it takes to get a subsample of 4. The
x-chart plots the individuals data directly.

On the other hand, the advantages of the average range calculation are:

1.  If there is an outlier in your data, the outlier will inflate the
standard deviation much less with the range calculation rather than the
sigma.  This is because the sigma calculation takes the square of the
distance of each datum from the average.  With the standard deviation
inflated less, the outler is easier to detect.  I will point out that some
people have suggested using the median range to eliminate this effect even
further.  In the case of using the sigma calculation, one must be very wary
to investigate points near the control limits, and do trial calculations
"throwing them out" from the sigma calculation to see if that puts them
outside the resulting control limits.

2.  A fixed subsample size (usually 4 or 5) is a more "rational" subgrouping
than accumulating by calendar month.  For example, one month's data may be
based upon 10 items, and the next month based upon 2.  This comes into play
if you are calculating the average value (such as average cycle time) for
the month.

One note:  If the data has come from a stable population, both the average
range and the sigma calculation should tend to the same answer as sample
size increases.  Try it yourself sometime with a random number generator
sometime.  Also, purposely throw in an outlier to see its effect on both
calculations.

For other questions in Dan's list - the cusum control chart is an entirely
different construction chart than an x-chart or xBar-R.  The allowing two
outliers every month based upon two standard deviation limits is a practice
I have not seen before, and I think one can document that neither Deming,
Shewhart, or Wheeler would support it.  

Steve Prevette
ESH Planning and Performance
Fluor Hanford, A Fluor Global Services Company
ASQ Certified Quality Engineer
steven_s_prevette@rl.gov
509-373-9371
=======================================================================