# Definition:Variance

## Definition

### 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.

## Also denoted as

The notation $\operatorname {\mathsf {var} } \sqbrk X$ can also be seen for $\var X$.

## Technical Note

The $\LaTeX$ code for \(\var {X}\) is `\var {X}`

.

When the argument is a single character, it is usual to omit the braces:

`\var X`

## Sources

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