On 2016-04-22 14:36:47 +0000, firstname.lastname@example.org said:
> Suppose you have some real valued measurements of something. > > (1) The mean minimizes the sum of squared deviations of each > observation from it. > > (2) The average of the largest and smallest observation minimizes the > maximum absolute value of the deviation from it.
Otherwise known as the midrange.
> (3) Do other measures of central tendency such as the median have a > similar property?
The median minimizes the sum of absolute deviations. Otherwise known as the L_1 fit.
> (4) Is there a closed form formula for the point that minimizes the sum > of the absolute value of the deviations from it?
Oterwise known as the median.
Before you even ask, yes there are versions of regression for the various norms.