variance


Definition

The varianceMathworldPlanetmath of a real-valued random variableMathworldPlanetmath X is

Var⁑X=𝔼⁒[(X-m)2],m=𝔼⁒X,

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.

Usage

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.

Properties

  1. 1.

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

    Var⁑X=𝔼⁒[X2]-𝔼⁒[X]2.

    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:

    Var⁑X=infaβˆˆβ„β‘π”Όβ’[(X-a)2].
  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:

    Var⁑[a⁒X+b⁒Y]=a2⁒Var⁑X+b2⁒Var⁑Y+2⁒a⁒b⁒Cov⁑[X,Y],

    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:

    Var⁑Xβ‰ˆs2=1n-1β’βˆ‘i=1n(xi-xΒ―)2,xΒ―=1nβ’βˆ‘j=1nxj.
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