# How to prove that the variance about mean is min?

• October 25th 2007, 07:07 PM
chopet
How to prove that the variance about mean is min?
or equally, the mean is the central point which minimises the sum of squares of deviation.

Use least mean square regression?
• November 2nd 2007, 03:31 AM
chopet
hello. anyone here can provide me solutions or give me guidance?

Thanks.
• November 2nd 2007, 05:37 AM
CaptainBlack
Quote:

Originally Posted by chopet
or equally, the mean is the central point which minimises the sum of squares of deviation.

Use least mean square regression?

Do you mean this:

Show that the mean $\bar{x}$ is the value of $k$ which minimises the second moment of a RV $X$ about $k$?

That is $k=\bar{x}$ minimises:

$
\int (x-k)^2 p(x) ~dx
$

Differentiate wrt $k$ and solve:

$
\frac{d}{dk}f(k)=0
$

RonL