I assume "average" was supposed to say "arithmetic mean".
That's important because both the mean and the median
are averages. They use different notions of distance,
but that's all.
This becomes particularly clear in terms of the
Minkowski Lp norm. We define the distance from
some reference point (c) to be:
D(c) = ( sum |xi - c|^p )^(1/p)
i
and we find the average by minimizing D with respect
to c.
-- The median corresponds to p=1 (Manhattan metric).
-- The arithmetic mean corresponds to p=2 (Euclidean metric).
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If anybody isn't 100% familar with the idea of Lp norm,
I heartily recommend the following exercise: Let xi
be the components of a roving position vector in two
dimensions, and consider D(0) = 1, i.e. the "unit
circle" i.e. the locus of unit distance from the origin.
Draw the "circles" for various values of p. Particularly
interesting values of p include
p=1 Manhattan metric
p=2 Euclidean metric
p=∞ Chebyshev metric aka "max"