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 $0$.
Definitions
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.
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.
Missing definition: central_moment
Examples
Each cell is the $k$th ___ of the random variable in that row.
| Example | statistical moment | central_moment | Why |
|---|---|---|---|
Missing example: fair_die_first_moments | |||
Missing example: fair_die_second_moments | |||
Missing example: fair_die_third_moments | |||
Missing example: standard_normal_second_moments | |||
Missing example: normal_mean_3_second_moments | |||
Notes
- The second central moment is the variance, and the first central moment of every random variable with a mean is $0$.
- The moments of a data set, called sample moments, average the powers of the data values, and the sample central moments average the powers of the deviations from the sample mean.