How can we increase the second moment (moment of inertia) in a physical system?

[KFC]

Hi there,
I am reading a math book with a lot of examples on mechanical physics. I saw a math term about second moment. In wiki, it is said that second moment is just moment of inertia in physics and has definition as below

$\int(x-\mu)^2f(x)dx$

here $\mu$ is the average and $f(x)$ is the weight or probability. Let me rewrite this into the summation with uniform weight as follow

$\dfrac{\sum_i (x_i-\mu)^2}{N} = \langle x^2\rangle - \langle x\rangle^2$

I am trying to associate this with moment of inertia and try to figure out how can we increase the moment of inertia based on this formula. In that formula, if we want to increase the second moment, we should decrease the average of $x$, if $x$ is mass, does it mean that I need to put all mass pieces as close to others as possible so to have minimum average? If I did that, how can I tell that won't decrease the first term as well? Thanks.

 

[mathman]

Your best bet is to increase \langle x^2\rangle while keeping the mean constant. This means moving things away from the mean on opposite sides.

 

[andrewkirk]

The moment of inertia is like the variance of a set of statistical data. In fact, if all particles have equal weight, it is proportional to the sum of the variances of the x,y and z coordinates of the particles in the body.

The variance is the second central moment of a distribution. It is increased by moving the data further away from one another. In the physical case, that means moving the particles further away from one another.

One way to keep the mean constant is to just make some changes that move the particles further away from one another, then measure the mean and then translate all the particles in parallel to put the mean back where it was. Parallel translations do not change the moment of inertia (second central moment).