Definition:Dispersion (Statistics)

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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:


Range

Definition:Range (Statistics)

Interquartile Range

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

The interquartile range is defined and denoted as:

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


Mean Absolute Deviation

Definition:Mean Absolute Deviation

Variance

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