# 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:

### Range

### 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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**dispersion**

- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel:
*Time Series Analysis: Forecasting and Control*(3rd ed.) ... (previous) ... (next):

- 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

- $2.1$ Autocorrelation Properties of Stationary Models:

- $2$: Autocorrelation Function and Spectrum of Stationary Processes:

- Part $\text {I}$: Stochastic Models and their Forecasting:

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**dispersion** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**measure of dispersion**