Minimum Value of Absolute Deviation

[maverick280857]

Hi,

How can I rigorously prove that the quantity

$$S = \sum_{i=1}^{n}|X_{i} - a|$$

(where $X_{1},\ldots,X_{n}$ is a random sample and a is some real number) is minimum when a is the median of the $X_{i}$'s?

Thanks.

 

[maverick280857]

Ok I think I got it. If a is the median, then there are just as many numbers less than it than there are greater than it...but how do I write the median in terms of the random sample?

 

[quadraphonics]

Ok I think I got it. If a is the median, then there are just as many numbers less than it than there are greater than it...but how do I write the median in terms of the random sample?

There's no simple way to write it like you can with, say, the mean. What you can do is re-order the sample in increasing order, such that $X_1 \leq \ldots \leq X_{n/2} \leq a \leq X_{n/2 + 1} \leq \ldots \leq X_n$. For the actual proof, you might try working by contradiction: assume some other value of a results in the lowest value for the sum, and then show that you can construct an even lower value by moving a towards the median.