# Sum of deviations from median

1. Dec 26, 2014

### gianeshwar

Dear Friends!
Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?

2. Dec 26, 2014

### Stephen Tashi

I assume you're talking about the sum of the deviations of the values in a sample from a statistic computed from a sample. The sum of the deviations of sample values from the sample mean is zero.

3. Dec 27, 2014

A median minimizes this sum (in a sample) as a function of $a$
$$\sum_{i=1}^n |x_i - a|$$

The mean (sample average if you will) minimizes this sum as a function of $a$
$$\sum_{i=1}^n \left(x_i - a\right)^2$$

I don't know of any minimizing property for the mode.

4. Dec 27, 2014

### pwsnafu

5. Dec 29, 2014