Central Moment

probability-theory

Definition

Central Moment

A central moment is calculated about the mean $\mu =\mathbb{E}[X]$ of the distribution. The $k$-th central moment describes the shape of the distribution relative to its center.

The $k$-th central moment is given by:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^k]$$

Note that for $k=1$, the central moment is:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^1]=\mathbb{E}[X]-\mathbb{E}[\mathbb{E}[X]]=\mathbb{E}[X]-\mathbb{E}[X]=0$$

Variance

For $k=2$, the second central moment is:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^2]$$

which is the exact definition of the variance.

Intuition

After finding the centre, the next logical question is: “How spread out are the data points from this centre?“. Are they all clustered tightly around the mean, or are they widely dispersed.

To measure this, one could try calculating the average distance of each point from the mean, i.e.:

$$\mathbb{E}[X-\mathbb{E}[X]]$$

However, some points are to the left mean (negative distance) and some are to the right (positive distance). These will cancel each other out, and the result will always be zero.

Thus, all distances should be non-negative. There are two obvious main options:

1. Absolute value: $\mathbb{E}[\left|X-\mathbb{E}[X]\right|]$
2. Squaring: $\mathbb{E}[{\left(X-\mathbb{E}[X]\right)}^2]$

Statisticians prefer squaring for a few reasons: it’s mathematically easier to work with (continuous, differentiable), and it heavily penalises values that are far from the mean, which is often a desirable property.

This measure, the average of the squared distances from the mean, is exactly the variance:

$$\begin{array}{rl}\mathbb{V}ar(X)=\mathbb{E}[(X-\mathbb{E}[X]{)}^2]& =\text{2nd Central Moment}\\ & =\text{Spread or Dispersion}\end{array}$$