In mathematics, the variance of a realvalued random variable is its second central moment. Hence for random samples x_{i} where i=1, 2, ..., the variance σ^{2} is
\left( x_i  \overline{x} \right) ^ 2</math>
So the variance of a set of data is the mean squared deviation from the arithmetic mean of the same set of data. Because this calculation sums the squared deviations, we can conclude two things:
One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum of independent random variables is the sum of their variances. (A weaker condition than independence, called "uncorrelatedness" also suffices.)
If X is a vectorvalued random variable, with values in R^{n}, and thought of as a column vector, then its variance is E((Xμ)(Xμ)'), where μ=E(X) and X' is the transpose of X, and so is a row vector. The variance is a nonnegativedefinite square matrix.
If X is a complexvalued random variable, then its variance is E((Xμ)(Xμ)^{*}), where X^{*} is the complex conjugate of X. This variance is a nonnegative real number.
When the set of data is a population, we call this the population variance. If the set is a sample, we call it the sample variance. When estimating the population variance of a finite sample, the following formula gives an unbiased estimate:
See algorithms for calculating variance.
See also: standard deviation, arithmetic mean, skewness, kurtosis, statistical dispersion
In land use, a variance is an administrative exception to zoning regulations, generally in order to cure a deficiency in a real property which would prevent the property from complying with the zoning regulation.
An example: suppose a "low density residential" zone requires that a house have a setback (the distance from the edge of the property to the edge of the building) of no less than 100 feet. If a particular property were only 100 feet deep, it would be impossible to build a house on the property, potentially resulting in an unlawful regulatory taking. A variance exempting the property from the setback regulation would allow a house to be built.
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