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Variance Metric: Difference between revisions
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* http://en.wikipedia.org/wiki/Variance#Definition | * http://en.wikipedia.org/wiki/Variance#Definition | ||
** QUOTE: If a [[random variable]] ''X</i> has the [[expected value]] (mean) {{nowrap|1 = ''μ'' = E[''X'']}}, then the variance of ''X</i> is given by: :<math>\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]. \,</math> <P> That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. This definition encompasses random variables that are [[discrete random variable|discrete]], [[continuous random variable|continuous]], or neither (or mixed). It can be expanded as follows: :<math>\begin{align} \operatorname{Var}(X) &= \operatorname{E}\left[(X - \mu)^2 \right] \\ &= \operatorname{E}\left[X^2 - 2\mu X + \mu^2 \right] \\ &= \operatorname{E}\left[X^2 \right] - 2\mu\,\operatorname{E}[X] + \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - 2\mu^2 + \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2. \end{align}</math> <P> A mnemonic for the above expression is "mean of square minus square of mean". <P> The variance of random variable ''X</i> is typically designated as Var(''X''), <math>\scriptstyle\sigma_X^2</math>, or simply σ<sup>2</sup> (pronounced | ** QUOTE: If a [[random variable]] ''X</i> has the [[expected value]] (mean) {{nowrap|1 = ''μ'' = E[''X'']}}, then the variance of ''X</i> is given by: :<math>\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]. \,</math> <P> That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. This definition encompasses random variables that are [[discrete random variable|discrete]], [[continuous random variable|continuous]], or neither (or mixed). It can be expanded as follows: :<math>\begin{align} \operatorname{Var}(X) &= \operatorname{E}\left[(X - \mu)^2 \right] \\ &= \operatorname{E}\left[X^2 - 2\mu X + \mu^2 \right] \\ &= \operatorname{E}\left[X^2 \right] - 2\mu\,\operatorname{E}[X] + \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - 2\mu^2 + \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2. \end{align}</math> <P> A mnemonic for the above expression is "mean of square minus square of mean". <P> The variance of random variable ''X</i> is typically designated as Var(''X''), <math>\scriptstyle\sigma_X^2</math>, or simply σ<sup>2</sup> (pronounced “[[sigma]] squared"). | ||
=== 2005 === | === 2005 === | ||
* ([[Lord et al., 2005]]) ⇒ [[Dominique Lord]], [[Simon P. Washington]], and [[John N. Ivan]]. ([[2005]]). | * ([[Lord et al., 2005]]) ⇒ [[Dominique Lord]], [[Simon P. Washington]], and [[John N. Ivan]]. ([[2005]]). “Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory.” In: Accident Analysis & Prevention, 37(1). [http://dx.doi.org/10.1016/j.aap.2004.02.004 doi:10.1016/j.aap.2004.02.004] | ||
** QUOTE: The [[arithmetic mean|mean]] and [[arithmetic variance|variance]] of the [[binomial distribution]] are <math>E(Z) = Np</math> and <math>VAR(Z) = Np(1-p)</math> respectively. | ** QUOTE: The [[arithmetic mean|mean]] and [[arithmetic variance|variance]] of the [[binomial distribution]] are <math>E(Z) = Np</math> and <math>VAR(Z) = Np(1-p)</math> respectively. | ||
=== 1987 === | === 1987 === | ||
* ([[Davidian & Carroll, 1987]]) ⇒ M. Davidian and R. J. Carroll. (1987). | * ([[Davidian & Carroll, 1987]]) ⇒ M. Davidian and R. J. Carroll. (1987). “Variance Function Estimation.” In: Journal of the American Statistical Association, 82(400). http://www.jstor.org/stable/2289384 | ||
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