In probability theory and statistics, variance is a measure of dispersion, meaning it is a measure of how far a set of numbers are spread out from their average value. It is defined as the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically, it is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by σ 2 {\displaystyle \sigma ^{2}} , s 2 {\displaystyle s^{2}} , Var ( X ) {\displaystyle \operatorname {Var} (X)} , V ( X ) {\displaystyle V(X)} , or V ( X ) {\displaystyle \mathbb {V} (X)} . 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 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 estimate the population variance on the basis of the sample 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 sampling.
Definition The variance of a random variable X {\displaystyle X} is the expected value of the squared deviation from the mean of X {\displaystyle X} , μ = E [ X ] {\displaystyle \mu =\operatorname {E} [X]} :
Var ( X ) = E [ ( X − μ ) 2 ] . {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right].}
This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:
Var ( X ) = Cov ( X , X ) . {\displaystyle \operatorname {Var} (X)=\operatorname {Cov} (X,X).}
The variance is also equivalent to the second cumulant of a probability distribution that generates X {\displaystyle X} . The variance is typically designated as Var ( X ) {\displaystyle \operatorname {Var} (X)} , or sometimes as V ( X ) {\displaystyle V(X)} or V ( X ) {\displaystyle \mathbb {V} (X)} , or symbolically as σ X 2 {\displaystyle \sigma _{X}^{2}} or simply σ 2 {\displaystyle \sigma ^{2}} (pronounced "sigma squared"). The expression for the variance can be expanded as follows:
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