From: bruck@math.usc.edu (Ronald Bruck)
Newsgroups: sci.math.num-analysis
Subject: Re: Mean, Median, Average ?
Date: 15 Jul 1998 17:33:14 -0700

In article <35acb260.0@news1.igateway.com>, Bean <thitt@igateway.com> wrote:
>Could someone explain the difference between
>Mean, Median, and Average ?

Mean (arithmetic mean) is the same as average; you take the sum of the
values, say a1, a2, ..., an, and divide by n:

  mean = (a1 + a2 + ... + an)/n.

Median is the number which has as many values greater than it as less than
it in the dataset; that is, it's smack-dab in the middle.  If the dataset
has an even number of members, it's usually taken to be the midpoint of the
two values at the very middle.

Finally:  although you didn't ask, "mode" is the MOST FREQUENT observation.

These quantities have the following variational interpretations:

Form the quantity f[p] = Abs[x-a1]^p + Abs[x-a2]^p + ... + Abs[x-an]^p,
and find a value xp which minimizes it.

  When p = 2, the minimizer is the mean.
  When p = 1, the minimizer is (a) median;
  When p = 0, this is the mode.

This last one has to be interpreted rather carefully; you should think of
it as letting xp be (a) minimizer of fp, and then take the limit of xp as
p decreases to 0.  (Otherwise we run into the problem of 0^0.)

When p = 1 and the dataset contains an even number of elements, then any
number between the two middle elements will minimize fp.  That's why I say
it's (a) median.

--Ron Bruck