Is the Sum of Deviations from the Median Always the Smallest?

[gianeshwar]

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

 

[Stephen Tashi]

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

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.

 

[statdad]

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.

 

[pwsnafu]

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

Wikipedia has your answer.

 

[statdad]

OK - I didn't know of any useful minimization property for the mode.

 

[gianeshwar]

Thank You Friends!