# 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 $E \paren X$, 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.