Central Moment

Shipping

array_central_moment

The n-th central moment E[(x-μ)ⁿ] of an array (2 = variance, 3 = skew, 4 = kurtosis)

Signature

Inputs

dataArrayrequired— Any array, flattened to 1-D before the moment is taken.

Outputs

momentScalar— The n-th central moment about the mean.

Parameters

Key	Type	Default	Notes
`order`	int	2	Order $n$ of the central moment $E[(x-\mu)^n]$. $2$ = variance, $3$ = (unstandardized) skew, $4$ = (unstandardized) kurtosis.

Description

Central Moment computes the $n$ -th moment of the data about its mean, $μ_{n} = E [(x - μ)^{n}]$ . The order parameter picks $n$ : order $2$ is the variance, order $3$ the (raw, unstandardized) skewness numerator, order $4$ the (raw) kurtosis numerator. Order $0$ is $1$ by definition and order $1$ is $0$ (the first central moment always vanishes).

Unlike the standardized shape statistics on Array Summary, this node returns the raw moment in the data's units $^{n}$ — useful when you need the moment itself (e.g. to build a custom estimator) rather than a dimensionless ratio. The node is stateless.

Mathematics

μ_{n} = E [(x - μ)^{n}] = \frac{1}{N} i = 1 \sum N (x_{i} - μ)^{n}, μ = \frac{1}{N} i \sum x_{i}

Examples

Variance via order 2

With order = 2, the node returns $μ_{2} = σ^{2}$ . For $[1, 2, 3, 4, 5]$ (mean $3$ ) that is $\frac{1}{5} (4 + 1 + 0 + 1 + 4) = 2$ .

Applications

Building custom shape estimators from raw moments.
Feeding higher-order moments into method-of-moments parameter fits.
Teaching / diagnostics where the un-normalized moment is the quantity of interest.

Neat

Order is clamped to $\ge 0$ and truncated to an integer, so the node degrades gracefully for out-of-range inputs instead of erroring.

The first central moment is identically zero and the zeroth is one — a handy sanity check that the reduction is centred correctly.