How does that answer the question? It's true that the center of gravity is a mean, but the moment of inertia is not a variance. It's one thing to say something is "proportional to a variance" to mean that the constant is 2 or pi, but when the constant is the number of points, I think it's missing the statistical point.

But the bigger problem is that these are not statistical examples! Means and sums of squares occur many places, but why are they are a good choice for the central tendency and the tendency to be central? Are you suggesting that we think of a random variable as a physical rod? Why? Does trying to spin it have any probabilistic or statistical meaning?

Mean and variance are closely related to center of mass and moment of inertia. This is good intuition to have, and it's statistical. The only difference is that the first two are moments of a probability distribution, and the second two are moments of a mass distribution.

2solipsist6yI wasn't aiming to answer Locaha's question as much as figure out what question
to answer. The range of math knowledge here is high, and I don't know where
Locaha stands. I mean,
That could be a basic question about the meaning of averages -- the sort of
knowledge I internalized so deeply that I have trouble forming it into words.
But maybe Locaha's asking a question like:
That's a less philosophical question. So if Locaha says "means are like the
centers of mass! I never understood that intuition until now!", I'll have a
different follow up than if Locaha says "Yes, captain obvious, of course means
are like centers of mass. I'm asking about XYZ".

How does that answer the question?

It's true that the center of gravity is a mean, but the moment of inertia is not a variance. It's one thing to say something is "proportional to a variance" to mean that the constant is 2 or pi, but when the constant is the number of points, I think it's missing the statistical point.

But the bigger problem is that these are not statistical examples! Means and sums of squares occur many places, but why are they are a good choice for the central tendency and the tendency to be central? Are you suggesting that we think of a random variable as a physical rod? Why? Does trying to spin it have any probabilistic or statistical meaning?

Mean and variance are closely related to center of mass and moment of inertia. This is good intuition to have, and it's statistical. The only difference is that the first two are moments of a probability distribution, and the second two are moments of a mass distribution.