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In mathematics, the variance of a real-valued random variable is its second central moment. Hence for random samples xi where i=1, 2, ..., the variance σ2 is

<math>\sigma^2 = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n
 \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:

  1. The variance is never negative because the squares are positive or zero. When any method of calculating the variance results in a negative number, we know that there has been an error, often due to a poor choice of algorithm.
  2. The unit of variance is the square of the unit of observation. Thus, the variance of a set of heights measured in inches will be given in square inches. This fact is inconvenient and has motivated statisticians to call the square root of the variance, the standard deviation and to quote this value as a summary of dispersion.

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 vector-valued random variable, with values in Rn, and thought of as a column vector, then its variance is E((X-μ)(X-μ)'), where μ=E(X) and X' is the transpose of X, and so is a row vector. The variance is a nonnegative-definite square matrix.

If X is a complex-valued random variable, then its variance is E((X-μ)(X-μ)*), 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:

<math>s^2 = \frac{1}{n-1} \sum_{i=1}^n \left( x_i - \overline{x} \right) ^ 2</math>

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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