From: Ray Hicks (rh208@cus.cam.ac.uk)
Date: Thu Mar 19 1998 - 14:46:56 EST
Hi Marcus,
The arithmetic mean is the "best" measure of central tendency for true
normal (gaussian) distributions, the geometric mean is the same for
distributions that are made normal if they are transformed logarithmically
(log-normal distribution).
The mean is calculated by summing the values of all measurements and then
dividing by the number (n) of measurements, whereas the geometric mean is
found by multiplying all measurements and finding the nth root of the
product. The geometric mean sounds hard to calculate, but is what you get
if you antilog the arithmetic mean of log data, hence its widespread
appearance in flow cytometric analysis programs. If a "log" sample is
normally distributed (ie Log-normal), then the geometric mean would
indicate the centre better than the arithmetic mean.
The median (50th centile) is the value that corresponds to the middle item
in a ranked list (ie sorted by magnitude) of all measurements. It's robust
in that it doesn't necessarily move in response to small numbers of
outliers, or to skewing of the tails of a distribution, whereas the mean is
tugged by both. One situation where the median is probably the only valid
measure is where data pile up at one extreme of measurement, as long as
more than 50% of the cells are clear of the sides you get a valid median,
but either type of mean will be way off.
So if you've got normally distributed data, and want to be able to reflect
small changes, use the arithmetic mean, if you've got lognormal data use
the geometric mean, and if you want a robust indicator use the median (but
this will fail to indicate subtle changes).
A great deal of flow cytometric data seems to skew off to the right, and
looks pretty normal when logged, but I think you'd have to use a test like
Kolmogorov-Smirnov to test your distribution against normal and lognormal
and plump for arithmetic or geometric means depending on the outcome.
There might be better tests, and if there are I'm sure that they'll be
pointed out soon enough, in the meantime I'll be adding this function to
FCS Assistant or FCSPress.
Ray
At 8:46 +0000 19/3/98, Marcus Reckermann wrote:
>Hi flow folks,
>I have a question on basic statistics, and I hope its not too stupid:
>I´m wondering what to use: "mean" or "geometric mean" as a relative
>fluorescence measure; and how about the "median"?? I use the
>linearized data from CellQuest (not the channel numbers), but I
>analyse in WinMDI because from there I can paste the data direct into
>Excel. Thanks for any hints in advance.
>marcus
>__________________________________________________
>Dr. Marcus Reckermann
>Research- and Technology Centre Westcoast (FTZ)
>Hafentoern
>D-25761 Buesum
>Germany
>Tel: +49-4834-604-204 or -261
>Fax: +49-4834-604-299
>E-Mail: recker@ftz-west.uni-kiel.de
Ray Hicks
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