# variance

## Definition

The variance of a real-valued random variable $X$ is

 $\operatorname{Var}X=\mathbb{E}\bigl{[}(X-m)^{2}\bigr{]}\,,\quad m=\mathbb{E}X\,,$

provided that both expectations $\mathbb{E}X$ and $\mathbb{E}[(X-m)^{2}]$ exist.

The variance of $X$ is often denoted by $\sigma^{2}(X)$, $\sigma^{2}_{X}$, or simply $\sigma^{2}$. The exponent on $\sigma$ is put there so that the number $\sigma=\sqrt{\sigma^{2}}$ is measured in the same units as the random variable $X$ itself.

The quantity $\sigma=\sqrt{\operatorname{Var}X}$ is called the standard deviation of $X$; because of the compatibility of the measuring units, standard deviation is usually the quantity that is quoted to describe an emprical probability distribution, rather than the variance.

## Usage

The variance is a measure of the dispersion or variation of a random variable about its mean $m$.

It is not always the best measure of dispersion for all random variables, but compared to other measures, such as the absolute mean deviation, $\mathbb{E}[\lvert X-m\rvert]$, the variance is the most tractable analytically.

The variance is closely related to the $\mathbf{L}^{2}$ norm for random variables over a probability space.

## Properties

1. 1.

The variance of $X$ is the second moment of $X$ minus the square of the first moment:

 $\operatorname{Var}X=\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}\,.$

This formula is often used to calculate variance analytically.

2. 2.

Variance is not a linear function. It scales quadratically, and is not affected by shifts in the mean of the distribution:

 $\operatorname{Var}[aX+b]=a^{2}\operatorname{Var}X\,,\quad\text{ for any a,b% \in\mathbb{R}.}$
3. 3.

A random variable $X$ is constant almost surely if and only if $\operatorname{Var}X=0$.

4. 4.

The variance can also be characterized as the minimum of expected squared deviation of a random variable from any point:

 $\operatorname{Var}X=\inf_{a\in\mathbb{R}}\mathbb{E}[(X-a)^{2}]\,.$
5. 5.

For any two random variables $X$ and $Y$ whose variances exist, the variance of the linear combination $aX+bY$ can be expressed in terms of their covariance:

 $\operatorname{Var}[aX+bY]=a^{2}\operatorname{Var}X+b^{2}\operatorname{Var}Y+2% ab\operatorname{Cov}[X,Y]\,,$

where $\operatorname{Cov}[X,Y]=\mathbb{E}[(X-\mathbb{E}X)(Y-\mathbb{E}Y)]$, and $a,b\in\mathbb{R}$.

6. 6.

For a random variable $X$, with actual observations $x_{1},\ldots,x_{n}$, its variance is often estimated empirically with the sample variance:

 $\operatorname{Var}X\approx s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2% }\,,\quad\bar{x}=\frac{1}{n}\sum_{j=1}^{n}x_{j}\,.$
 Title variance Canonical name Variance Date of creation 2013-03-22 11:53:46 Last modified on 2013-03-22 11:53:46 Owner stevecheng (10074) Last modified by stevecheng (10074) Numerical id 14 Author stevecheng (10074) Entry type Definition Classification msc 62-00 Classification msc 60-00 Classification msc 81-00 Classification msc 83-00 Classification msc 82-00 Classification msc 55-00 Related topic GeometricDistribution2 Related topic StandardDeviation Related topic Covariance Related topic MeanSquareDeviation