Editing Variance (section)

{{Short description|Statistical measure of how far values spread from their average}} 
{{About|the mathematical concept|other uses|Variance (disambiguation)}}
[[File:Comparison standard deviations.svg|thumb|400px|right|Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.]]

In [[probability theory]] and [[statistics]], '''variance''' is the [[expected value]] of the [[squared deviations from the mean|squared deviation from the mean]] of a [[random variable]]. The [[standard deviation]] (SD) is obtained as the square root of the variance. Variance is a measure of [[statistical dispersion|dispersion]], meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second [[central moment]] of a [[probability distribution|distribution]], and the [[covariance]] of the random variable with itself, and it is often represented by <math>\sigma^2</math>, <math>s^2</math>, <math>\operatorname{Var}(X)</math>, <math>V(X)</math>, or <math>\mathbb{V}(X)</math>.<ref>{{cite book |last1=Wasserman |first1=Larry |title=All of Statistics: a concise course in statistical inference |date=2005 |publisher=Springer texts in statistics |isbn=978-1-4419-2322-6 |page=51}}</ref>

An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the [[Average absolute deviation|expected absolute deviation]]; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.

There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical [[probability distribution]] and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.

The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include [[descriptive statistics]], [[statistical inference]], [[hypothesis testing]], [[goodness of fit]], and [[Monte Carlo method|Monte Carlo sampling]].

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[[File:variance_visualisation.svg|thumb|Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9): {{ordered list
    |A frequency distribution is constructed.
    |The centroid of the distribution gives its mean.
    |A square with sides equal to the difference of each value from the mean is formed for each value.
    |Arranging the squares into a rectangle with one side equal to the number of values, ''n'', results in the other side being the distribution's variance, ''σ''<sup>2</sup>.
}}]]