Relation between 3 STD and approximated STD from XmR

[Kromkowski@aol.com]

Paul Hollingworth correctly notes:
>>Incidentally one should take care to check how limits for 
Individuals charts are calculated in PC packages. I know of 
one (otherwise excellent) statistical package that offers 
individuals charts with limits drawn at 3 standard 
deviations (calculated as a single pass through the x values). This is NOT an XmR chart!<<

However, I'd be interested in commentary about the relationship between STD and the approximation of STD from XmR chart.

Generically, we say that the standard deviation (STD) of a 
population is the square root of the average squared 
deviation of the scores of the entire population from the 
mean.

However, we seldom can know the entire population which 
includes all indivuals, not just a sample, includes all those in the past as well as in the future.

Because we have to use a sample, we have agreed upon a kind 
of finigling factor, for the standard deviation of the 
sample where we use N-1.  Then through a kind of sleight of 
hand we say that STDS is useful (which it is) for saying 
things about future and past samples of the population. And 
as N approaches infinity, N-1 tends to approach N, the 
difference between infinity and infinity minus 1 is pretty 
doggone small.

The XmR is analogously making the same kind of finigle 
adjustment for the fact that we have a sample and do not 
nor cannot generally know ALL of the population (for the 
simplest reason we acknowlege that we can't know the 
future and for practical reasons, even if we could know the 
future, it would be a really difficult to use the whole 
population which might be infinite or sufficiently large 
to be a pain in the ass).

So for the XmR the finigle is that one STD approximately 
equals 1/d2.

What the XmR also claims is that its approximation tells us 
something useful about the relationship between consecutive 
samples.  But what is the basis of this claim:  Experience -
- Theory - Mathematics - a bit of each?

In addition the XmR also claims a superiority in 
practicality, in other words, it's a whole lot easier and 
faster (at least when done by hand) to calculate than the 
STDS because you don't have to use square-roots or make 
lots of small calculations -- the averaging, the 
subtracting, the summing, the dividing. (By the way, when 
done by hand each mathematical operation in the STDS 
formula creates an opportunity for a mistake.)

So when all is said and done, what, if any, mathematical or 
empirical relationship is there between the standard 
deviation of a sample and 1/d2?

If the advance of computers has taken away the praticality 
advantage, what is the current basis for use? And upon what theory is it based?

Or does it just work? Or if it was good enough for my ancestors then it is good enough for us?

P.S. I don't intend to stop using it any time soon, but I am open to a healthy debate.

JDK

Message posting through the Clemson CQI Web Server.