# Re: Idempotence and "Replication Insensitivity" are equivalent ?

Date: 25 Sep 2006 04:20:59 -0700

Message-ID: <1159183258.955436.16850_at_m7g2000cwm.googlegroups.com>

> My point sumply was that neither the minimizing property, nor any

*> other median definition defines the median uniquely in certain cases.
**>
*

Let me clarify this point.

In the case of the mean (AVG), the fact that it is a minimum for the sum of square deviation is necessary and sufficient. So this property, for the mean, is *equivalent* to the definition. (This happens because we are minimizing a convex function.)

In the case we are dealing with finite set of numbers, the median has of course the property that it minimizes the sum of absolute deviation, because it is contained in all the possible intervals x(i) - x(n-i+1), however, there may be multiple solution. We have multiple solution when n is even. So we would have:

median definition ==> minimumproperty

but not the other way round (ie., not equivalence)

In order to have a unique solution (and therefore equivalence), in the case we are dealing with finite set of numbers when the cardinality is not odd, the definition of MEDIAN is "completed" by *assuming* (it's an ASSUMPTION, a convention) as median the average of the 2 central terms of the ordered sequence of numbers:

MEDIAN = ( x(n\2) + x(n\2 + 1) ) / 2

for n odd there is no problem, as the median is the central term x((n+1)\2) .

For qualitative data is a different story...

-P Received on Mon Sep 25 2006 - 13:20:59 CEST