Moment

A moment is a specific quantitative measure of the shape of a function.

The \(n\)-th moment of a continuous function \(f(x)\) of a real variable \(x\) about a value \(c\) is

\[\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\diff x\]

Mean (\(n = 1\))

discrete:

\[\mu_1 =\sum x_i \cdot P(x)\]

with result \(x\) and probability function \(P()\)

continuous:

\[\operatorname \mu_1 = \int _{\mathbb {R}} x \cdot f(x)\diff x\]

Variance (\(n = 2\))

discrete: \(\(\mu_2 =\sum x \cdot P(x)\)\)

Central Moments Comparison

Moment	Statistics	Mechanics
0.	Total P=1.0	mass
1.	mean	center of mass
2.	variance	rotational inertia
3.	skewness