Fellow Allstatters:
Thanks a lot to all those who answered my query on whether there is such
a thing as an algebraic mean. The correct answer (from Ivailo Parchev
and Robert Nemeth) appears to be that, given a variable X, it is the
p'th root of the mean of X**p, where p is a power. It includes the
arithmetic and harmonic means as special cases for p=1 and -1,
respectively.
In general, it is the parameter which we are measuring if we do a t-test
after a power transformation, just as the geometric mean is the
parameter which we are measuring if we do a t-test after a log
transformation. However, if we prefer confidence intervals to P-values,
as all good statisticians do, then the confidence interval for the
difference between two means after the power transformation cannot be
transformed back to a confidence interval for anything as simple as a
ratio. Presumably, this is why not many statisticians know about the
algebraic mean, whereas not many statisticians do not know about the
geometric mean. (It is probably significant that, on a UK-based email
list, the only two people who sent me the "correct" answer appear to be
from a continental tradition of mathematics, presumably going back to
Hermann Minkowski.)
Sean McGuigan tells me that there is yet another mean called the
geometic mean. This is defined as follows. Given a random variable X,
define A_0 as its arithmetic mean and G_0 as its geometric mean. Then
define the sequences A_n and G_n recursively so that A_(n+1) is the
arithmetic mean of A_n and G_n, whereas G_(n+1) is the geometric mean of
A_n and G_n. The geometic mean is then equal to the common limit (as n
tends to infinity) of A_n and G_n. Apparently, Gauss did some work on
it, but I do not know whether anybody these days calculates confidence
limits for it (and, if so, how).
Thanks again to all who answered.
Roger
--
Roger Newson
Imperial College Medical School
Chelsea and Westminster Hospital (Fourth Floor)
369 Fulham Road, London SW10 9NH, UK
Tel: 0181 746 8163 International +44 181 746 8163
Fax: 0181 746 8151 International +44 181 746 8151
Email: [log in to unmask]
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