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

From: <pamelafluente_at_libero.it>
Date: 24 Sep 2006 03:03:19 -0700

Phil Carmody ha scritto:
>
> > 2. what is the median of {0, 1} and why.
> >

> Useless. It depends on the domain you're working in.

the median of the set of values { x1, x2} does not depend "on the "domain" you're working in." Anyway what do mean by "domain"? We are working with finite sets.

It can depend on the comparer you use, in general. But not for n= 2. For n=2, it is always coincident with the mean.

If we want to discuss about something and you take the freedom to insult someone without reason, you should at least show that youk know what we are talking about, otherwise you may end up not making a good show of yourself.

Anyway, let me recall some basics:

For a discrete numerical set {X1, ... , Xn} ,

MEDIAN {X1, ... , Xn}

• x ( (n+1)\2 ) , if n is odd (n = 2i+1, i=0,1,...)
• ( x ( n\2 ) + x ( n\2 + 1 ) ) / 2 , if n is even (n = 2i, i= 1,2,...)

where by x(i), I mean the i-th order statistic,  ie the i-th value in the ordered value (ordered according  to any custom comparer).

So the median does have an optimality property similar to the  mean. But in order to make this property a "characterizing" one,  the assumption is made that, for even cardinalities, the  average of the 2 (n>1) central terms is to be taken. In fact,  as Phil noted - in case n is even - all points between the 2  central terms (end point included) are point of minimum for the  sum of absolute deviations, as clearly they are contained in all  the intervals of type x(i) - x(n-i+1), i=0,...,n\2.

[ For categorical (say, "nonnumerical") data, the definition needs slight adjustments. ]

```  MEDIAN {0, 1} = .5
MEDIAN {0, 2, 1} = 1
MEDIAN {0, 1, 3, 2} = 1.5
MEDIAN {0, 4, 3, 1, 2} = 2

```

in the above cases MEDIAN() and AVG() are coincident   due to the "symmetry" of the finite support of the   discrete uniform distribution.

For n=1,2 or any case of symmetric support,   AVG and MEDIAN are always coincident,

i.e., MEDIAN {X1, X2} = AVG {X1, X2} for all X1, X2.

(For symmetric supports AVG() and MEDIAN() carry   the same information.)

The above concept of MEDIAN gets easily generalized to   the concept of QUANTILE [eg, quartile, percentile,...] Received on Sun Sep 24 2006 - 05:03:19 CDT

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