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

Range is a measure of dispersion of a set of observations.


Let $S$ be a set of observations of a quantitative variable.

The range of $S$ is defined as:

$\map R S := \map \max S - \map \min S$

where $\map \max S$ and $\map \min S$ are the greatest value of $S$ and the least value of $S$ respectively.


Interquartile Range

The interquartile range is a measure of dispersion in statistics.


Let $Q_1$ and $Q_3$ be first quartile and third quartile respectively.

The interquartile range is defined and denoted as:

$\operatorname {IQR} := Q_3 - Q_1$


Semi-Interquartile Range

The semi-interquartile range is a measure of dispersion in statistics.


Let $Q_1$ and $Q_3$ be first quartile and third quartile respectively.

The semi-interquartile range is defined as:

$\dfrac {Q_3 - Q_1} 2$

That is, half the interquartile range.


Mean Absolute Deviation

Let $S = \set {x_1, x_2, \ldots, x_n}$ be a set of observations.

Let $\bar x$ denote a measure of central tendency of $S$.


The mean absolute deviation with respect to $\bar x$ of $S$ is defined as the arithmetic mean of the absolute values of the deviation of the elements of $S$ from $\bar x$ :

$\ds \sum_{i \mathop = 1}^n \dfrac 1 n \size {x_i - \bar x}$


Variance

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

Standard deviation is a measure of dispersion of a set of observations.

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.


Also see

  • Results about dispersion in the context of statistics can be found here.


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
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