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Re: Idempotence and "Replication Insensitivity" are equivalent ?

From: vc <boston103_at_hotmail.com>
Date: 24 Sep 2006 19:36:12 -0700

Chris Smith wrote:
> Brian Selzer <brian_at_selzer-software.com> wrote:
[..]
> > You wrote, "There are some sets, such as {0, 1}, where every value between 0
> > and 1 (including both endpoints) is minimum."
> >
> > Unless 0 and 1 belong to some domain other than integers, whole numbers or
> > real numbers, it is clear that 0 is the minimum value of the set {0, 1}. I
> > don't know where you came up with the idea that both values are minimum.
>
> That statement was made, though, in the context of defining the median.
> The definition put forth (I don't recall by whom) is that the median is
> the number c such that the sum of the distances of each member of the
> set from c is minimized. In that context, the statement makes sense.
> When considering the set {0, 1}, any real number c from zero to one
> minimizes the sum of distances of members of the set from from c.

The veracity of the statement depends on *what* the set represents and what kind of median is discussed. In various contexts, the median can be either 0 or 1 or both, or the entire [0,1] interval(or 0.5 depending on the accepted convention).

In the case when any point in [0..1] can be a median, both the absolute deviation minimizing property and the more traditional definition would yield the same result which means that the OP (Pamela) was right and her opponent wrong: "I'm saying that the property does not always uniquely define a median ("*the* value", emphasis mine), and therefore cannot be used as the definition therefor."

He failed to produce a unique median definition which is not surprising since in certain circumstances no definition would yield a unique result. Received on Sun Sep 24 2006 - 21:36:12 CDT

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