The varianceMathworldPlanetmath of a real-valued random variableMathworldPlanetmath X is


provided that both expectations 𝔼⁒X and 𝔼⁒[(X-m)2] exist.

The variance of X is often denoted by Οƒ2⁒(X), ΟƒX2, or simply Οƒ2. The exponent on Οƒ is put there so that the number Οƒ=Οƒ2 is measured in the same units as the random variable X itself.

The quantity Οƒ=Var⁑X is called the standard deviationMathworldPlanetmath of X; because of the compatibility of the measuring units, standard deviation is usually the quantity that is quoted to describe an emprical probability distribution, rather than the variance.


The variance is a measure of the dispersion or variation of a random variable about its mean m.

It is not always the best measure of dispersion for all random variables, but compared to other measures, such as the absolute mean deviation, 𝔼⁒[|X-m|], the variance is the most tractable analytically.

The variance is closely related to the 𝐋2 norm for random variables over a probability space.


  1. 1.

    The variance of X is the second moment of X minus the square of the first moment:


    This formulaMathworldPlanetmathPlanetmath is often used to calculate variance analytically.

  2. 2.

    Variance is not a linear function. It scales quadratically, and is not affected by shifts in the mean of the distributionPlanetmathPlanetmath:

    Var⁑[a⁒X+b]=a2⁒Var⁑X,Β for anyΒ a,bβˆˆβ„.
  3. 3.

    A random variable X is constant almost surely if and only if Var⁑X=0.

  4. 4.

    The variance can also be characterized as the minimum of expected squared deviation of a random variable from any point:

  5. 5.

    For any two random variables X and Y whose variances exist, the variance of the linear combinationMathworldPlanetmath a⁒X+b⁒Y can be expressed in terms of their covarianceMathworldPlanetmath:


    where Cov⁑[X,Y]=𝔼⁒[(X-𝔼⁒X)⁒(Y-𝔼⁒Y)], and a,bβˆˆβ„.

  6. 6.

    For a random variable X, with actual observations x1,…,xn, its variance is often estimated empirically with the sample varianceMathworldPlanetmath:

Title variance
Canonical name Variance
Date of creation 2013-03-22 11:53:46
Last modified on 2013-03-22 11:53:46
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 14
Author stevecheng (10074)
Entry type Definition
Classification msc 62-00
Classification msc 60-00
Classification msc 81-00
Classification msc 83-00
Classification msc 82-00
Classification msc 55-00
Related topic GeometricDistribution2
Related topic StandardDeviation
Related topic Covariance
Related topic MeanSquareDeviation