Moments – Statistics’ physics connection

Interesting Fact:

Imagine you sculpted your distribution out of wood and tried to balance it on your finger. The balance point would be the mean regardless of the shape of the distribution !!

Now why is that?

This is where the concept of Moments come into play. Many of you might be wondering, how did the physics concept of ‘Moments’ came into statistics.

Well, statistics has a physics connection. 😅

Moments are the expected value of a Random variable. Moments define the characteristics and shape of a probability distribution.

Typically Moments about origin are called ‘Raw Moments’ and those around the mean are called ‘Central Moments’.

In fact, Mean is the first raw moment.

But Why the name moment ?

It turns out that statisticians named it so as an allusion to ‘Moment of inertia’.

Now coming back to our original question of ‘why the balance point of any distribution regardless of its shape is the mean’.

The first moment is the ‘center of mass’ of a probability distribution.

In physics, the center of mass is the point at which all the torques sume to zero.

A torque is a vector quantity which means that it has both a magnitude (size) and a direction associated with it.

If the size and direction of the torques acting on an object are exactly balanced, then there is no net torque acting on the object and the object is said to be in equilibrium.

Because there is no net torque acting on an object in equilibrium, an object at rest will stay at rest, and an object in constant angular motion will stay in angular motion.

Now you know why the balance point of any distribution is its mean.

I would urge the readers to read the excellent article on Understanding moments by Gregory Gundersen. Link in resources.
Also a link to drive home the intuition on torque is also in resources.

Resources:

Understanding moments – http://gregorygundersen.com/blog/2020/04/11/moments/