A median of x1, ..., xn is the middle value of x1, ..., xn  when n is odd, and any value...

A median of x1, ..., xn is the middle value of x1, ..., xn  when n is odd, and any value between the two middle values of x1, ..., xn when n is even. For example, if x1<x2<....< x5, the median is x3. If x1 < x2<x3 < x4, a median is any value between x2 and x3, including x2 and x3.

Use Exercises 125 and 126 and mathematical induction to prove that the sum

 n ≥ 1, is minimized when a is equal to a median of x1, ..., xn .

If we repeat an experiment n times and observe the values x1, ..., xn, the sum (3.2.9) can be interpreted as a measure of the error in assuming that the correct value is a. This exercise shows that this error is minimized by choosing a to be a median of the values x1, ..., xn. The requested inductive argument is attributed to J. Lancaster.