# 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 $\var X$.

## Also see

• Results about variance can be found here.

## 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