## LESSWRONGLW

I too spent a few years with a similar desire to understand probability and statistics at a deeper level, but we might have been stuck on different things. Here's an explanation:

Suppose you have 37 numbers. Purchase a massless ruler and 37 identical weights. For each of your numbers, find the number on the ruler and glue a weight there. You now have a massless ruler with 37 weights glued onto it.

Now try to balance the ruler sideways on a spike sticking out of the ground. The mean of your numbers will be the point on the ruler where it balances.

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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?

5IlyaShpitser6yMoments of mass in physics is a good intro to moments in stats for people who like to visualize or "feel out" concepts concretely. Good post!
4solipsist6yA different level explanation, which may or may not be helpful: Read up on affine space [http://en.wikipedia.org/wiki/Affine_space], convex combinations [http://en.wikipedia.org/wiki/Convex_combination], and maybe this article about torsors [http://math.ucr.edu/home/baez/torsors.html]. If you are frustrated with hand waving in calculus, read a Real Analysis textbook. The magic words which explain how the heck you can have a probability distributions over real numbers is measure theory [http://en.wikipedia.org/wiki/Measure_(mathematics\]).