(This article assumes you to know what the "mean", median and variance of a distribution are. In case you could need a quick refresher on this topic, please have a look at Basic measures of descriptive statistics first.)
Let's assume we would cut a thick, but flat, wooden model from a distribution like this. The usual basic measures of descriptive statistics would represent real physical properties of it.
(I know, this looks amazingly realistic. Sadly I'm currently not available to accept design projects. ^_-)
The mean is the center of gravity in regards to the x-axis. So we could balance it on a rod aligned with the mean line.
Also, we could hang it on a thread at one point on the mean line, and it would not tilt. I think this would make a very nice crib mobile. :)
We cut our wooden distribution model in two parts along the median, both halves have the exact same weight.
We spin our model around its vertical axis. It spins around its mean if we don't attach it to something. And the moment of inertia for this rotation is equivalent to the variance of our distribution. So the variance is proportional to the torque needed for some angular acceleration.