RE: SPC Theory Question

[Rip Stauffer <rstauffer@bluefirepartners.com>]

Can you use non-time-ordered data in an XmR chart? Jean-Marie makes a great
point. Fundamentally, an XmR chart relies on time order, as ranges between
subsequent pairs are used to estimate a sigma value. Like all process
behavior charts, though, it tests for a lack of random behavior over time.
If you took a series of values and enumerated every possible order in which
they might appear, you could end up, in some cases, with very small average
moving ranges. Consider the case, for example, where twenty data are ordered
0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9. The average
moving range would be .474, the mean would be 4.5, and the UNPL and LNPL
would be 5.76 and 3.24. Sixteen of the values would be outside the limits.
Granted, this is an extreme and rare ordering of the data, but not out of
the realm of possibility; and it serves to illustrate Jean-Marie's point:
"there are a lot of time orders." 

This was kind of an interesting question, so I tested it using some data.

I used 100 draws from a bead bowl. I then drew a 100 percent sample from
that set of numbers 30 times, and calculated moving ranges, average moving
ranges and 3-sigma limits (using the standard 2.66 mR-bar). The 3-sigma
distances ranged from 7.899 to 9.941, which suggests that with as many as
100 data, you could not count on the three sigma limits to be useful for
deciding whether there are any...what?..."special causes? (I don't think
so)"..."outliers? (maybe)"...well, just for the sake of argument, let's call
them "signals."

With 200 draws and the same procedure, I got 3-sigma distances from 8.087 to
9.544, which would also be less than useful. I skipped ahead, used 1000, and
this time got a range from 8.491 to 8.941. This suggests that, if you have
somewhere near 1000 data, you might be able to randomize them, plug the
resulting numbers into an XmR chart, and be able to discern "signals"
outside of three-sigma limits. 

I believe that the real question is, "Why?" Why do that? Analytic studies
generally deal with data over time, you are trying to achieve some
predictability. If you walk into an organization, and they hand you 1000
non-time-ordered data, and ask you whether the process from which the data
came is in statistical control, you would have to say "I don't know." Often,
we are handed a mess at the beginning of a study, and we just have to make
the best of it. You could randomize the data, plug them into an XmR chart,
and see whether there are any "signals." If you had some, you might say, "I
still don't know, but this suggests that there might be some (or a lack of)
signals. Even if you had no signals outside the three-sigma limits, you
would not be able to conclude anything, because without the time order, you
can't use any other indicators. 

Would you use those limits? Maybe you could use them as trial limits until
you get enough good, new, time-ordered data to develop a good chart. After
all, if you have 1000 data in random order and your assumption is that the
process will produce an approximately random order over time within some
limits, it seems reasonable to assume that those limits would be fairly
close to what you would get given time-ordered data from a process in
statistical control. In any case, before you make any claim for a reasonable
degree of statistical control, you would have to get some more data. 

This was a fun question! I can't wait to see some of the other responses.

Best regards to all, 

Rip

Rip Stauffer, Senior Consultant
BlueFire Partners
1300 Fifth St. Towers, 150 So. Fifth St.
Minneapolis, MN 55402
612-344-1027
mailto:rstauffer@bluefirepartners.com
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