Statistical moment
High school
Consider a random quantity $X$. The moments of $X$ are the long-run averages of the powers of $X$: the first moment is the average of $X$, the second moment is the average of $X^2$, the third moment is the average of $X^3$, and so on.
Undergraduate (probability)
Consider a random variable $X$ and a positive integer $k$. The $k$th moment of $X$ is the expected value
\[\mathbb{E}[X^k].\]Properties
- The first moment of a random variable is the mean of the random variable.
- The second central moment of a random variable is the variance of the random variable.
- The variance is the second moment minus the square of the first moment: $\operatorname{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$.
- The third and fourth central moments, divided by $\sigma^3$ and $\sigma^4$, where $\sigma$ is the standard deviation, are the skewness and the kurtosis.
Notes
The name comes from physics: the first moment of a mass distribution locates the center of mass, and the second moment is the moment of inertia.
The moments of a data set $x_1, \dots, x_n$, called sample moments, are the averages $(x_1^k + \dots + x_n^k) / n$.
Distinctions involving this term
- Moment vs. central moment: In statistics, moments and central moments each summarize the shape of a probability distribution by averaging powers. The $k$th moment $\mathbb{E}[X^k]$ of a random variable $X$ averages powers of $X$ itself, measured from $0$. The $k$th central moment $\mathbb{E}[(X - \mu)^k]$ averages powers of the deviation of $X$ from its mean $\mu$, so central moments do not change when a constant is added to $X$. The moments and the central moments of $X$ agree when the mean of $X$ is $0$.