# Variance V

The variance is a measure of how much the probability distribution is spread from the mean:

V=E\bigg[ \big(x-\bar{x}\big)^2\bigg] = E\big( x^2\big)-\bar{x}^2 |

where denotes the expected value (mean) of whatever is in the brackets, so:

V=\int_{-\infty}^\infty \big(x-\bar{x}\big)^2 .f(x) .dx |

The variance sums up the squared distance from the mean of all possible values of *x*, weighted by the probability of *x* occurring. The variance is known as *the second moment about the mean*. It has units that are the square of the units of *x*. So, if *x* is cows in a random field, *V* has units of cows2. This limits the intuitive value of the variance.

Variance and standard deviation have the following properties, where *a* is some constant and X, Xi are random variables:

V\big(X\big)\geq 0 | and | \sigma\big(X\big)\geq 0 |

V\big(aX \big)=a^2V\big(X\big) | and | \sigma\big(aX\big)=a\sigma\big(X\big) |

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