measure of dispersion for geometric mean: geometric variance?

[Tim Peters]

Having never seen a benchmark program worth running, let alone worth
worrying about <grin>, maybe the gang would be more interested in ways
to make their current numbers appear better.

Seriously, the following apparently isn't known by many people who ought
to know it (including everyone who needs to supply fodder for their
local Mean Monster):

Given a fixed vector of positive data points x(1), x(2), ..., x(n),
define the "generalized mean" function gmean(p) as

   gmean(p) = ((sum for i = 1 to n of (x(i)**p)) / n) ** (1/p)

I.e., gmean(p) is just the p'th root of the average value of x**p.

Then

   minimum value in x(:)   = limit gmean(p) as p goes to -infinity
   harmonic   mean of x(:) = gmean(-1)
   geometric  mean of x(:) = limit gmean(p) as p goes to 0
   arithmetic mean of x(:) = gmean(1)
   maximum value in x(:)   = limit gmean(p) as p goes to +infinity

So long as the data points aren't identically equal, gmean(p) is
monotonically increasing, and yields an uncountable infinity of
"averages" we *could* be beating each other over the head with.  The
ones we usually fight about are just boring little special cases.

The marketing application is to do "gmean vs p" plots for you and your
competitor, then advertise your gmean(p) at some value of p for which
you're higher (the value of gmean(p) should be in large bold type; p
itself fits very nicely in an illegible footnote).  E.g., Cray these
days spends a lot of time advertising gmean(+infinity) <grin>.

A more interesting application is to do a "gmean vs p" plot for your own
benefit:  with a little practice, you'll find that the shape of the
curve says a *lot* of interesting stuff about the data.

but-also-tedious-to-incorporate-in-short-ascii-messages-hence-something-
   that-will-never-catch-on-ly y'rs  - tim


Tim Peters   Kendall Square Research Corp
timaksr.com,  ksr!timaharvard.harvard.edu