# Definition:Dispersion (Statistics)

## Definition

Let $S$ be a sample of a population in the context of statistics.

The dispersion of $S$ is a general term meaning how much the data describing the sample are spread out.

The word can also be applied to a random variable.

Measures of dispersion include the following:

### Interquartile Range

Let $Q_1$ and $Q_3$ be first and third quartiles.

The interquartile range is defined and denoted as:

$\operatorname {IQR} := Q_3 - Q_1$

### Discrete Random Variable

Let $X$ be a discrete random variable.

Then the variance of $X$, written $\var X$, is a measure of how much the values of $X$ varies from the expectation $\expect X$, and is defined as:

$\var X := \expect {\paren {X - \expect X}^2}$

That is: it is the expectation of the squares of the deviations from the expectation.

### Continuous Random Variable

Let $X$ be a continuous random variable.

Then the variance of $X$, written $\var X$, is a measure of how much the values of $X$ varies from the expectation $\expect X$, and is defined as:

$\var X := \expect {\paren {X - \expect X}^2}$

That is, the expectation of the squares of the deviations from the expectation.

### Standard Deviation

Let $X$ be a random variable.

Then the standard deviation of $X$, written $\sigma_X$ or $\sigma$, is defined as the principal square root of the variance of $X$:

$\sigma_X := \sqrt {\var X}$

## Also known as

Dispersion is also known as spread.

## Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.2$ Stationary Stochastic Processes: Mean and variance of a stationary process