The Average Puzzle

Suppose we have a set of N numbers, xi, and I want to choose an "average" value x*. There are many ways to define an average, but I have a particular requirement in mind. How should I choose the average value so that

  1. I get the smallest possible value for the sum of the absolute values of the differences |xi-x*|;
  2. I get the smallest possible value for the sum of the squares of the differences (xi-x*)2;
  3. I get the smallest possible value for the maximum of the absolute values of the differences |xi-x*|;

Thanks to Jim Fink for asking this question.

I give up, show me the solution.

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Last revised on 08 September 2005.