variance

Definition

The variance of a real-valued random variable $X$ is$$ \Var X = \E\bigl[ (X - m)^2 \bigr]\,, \quad m = \E X\,,$$ provided that both expectations $\E X$ and $\E[(X-m)^2]$ 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 $\sigma = \sqrt{\Var X}$ 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, $\E[ \abs{X-m} ]$ , the variance is the most tractable analytically.

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

Properties

1. The variance of $X$ is the second moment of $X$ minus the square of the first moment:$$ \Var X = \E[X^2] - \E[X]^2\,.$$ This formula is often used to calculate variance analytically.
2. Variance is not a linear function. It scales quadratically, and is not affected by shifts in the mean of the distribution:$$ \Var[ aX + b ] = a^2 \Var X\,, \quad \text{ for any $a, b \in \real$.}$$
3. A random variable $X$ is constant almost surely if and only if $\Var X = 0$ .
4. The variance can also be characterized as the minimum of expected squared deviation of a random variable from any point:$$ \Var X = \inf_{a \in \real} \E[(X-a)^2]\,.$$
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:$$ \Var[aX+bY] = a^2 \Var X + b^2 \Var Y + 2ab \Cov[X,Y]\,,$$ where $\Cov[X,Y] = \E[(X-\E X)(Y-\E Y)]$ , and $a, b \in \real$ .
6. For a random variable $X$ , with actual observations , its variance is often estimated empirically with the sample variance:$$ \Var X \approx s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2\,, \quad \bar{x} = \frac{1}{n} \sum_{j=1}^n x_j\,.$$