Editing Variance (section)

====Matrix notation for the variance of a linear combination====

Define <math>X</math> as a column vector of <math>n</math> random variables <math>X_1, \ldots,X_n</math>, and <math>c</math> as a column vector of <math>n</math> scalars <math>c_1, \ldots,c_n</math>. Therefore, <math>c^\mathsf{T} X</math> is a [[linear combination]] of these random variables, where <math>c^\mathsf{T}</math> denotes the [[transpose]] of <math>c</math>. Also let <math>\Sigma</math> be the [[covariance matrix]] of <math>X</math>. The variance of <math>c^\mathsf{T}X</math> is then given by:<ref>{{Cite book | last1=Johnson | first1=Richard | last2=Wichern | first2=Dean | year=2001 | title=Applied Multivariate Statistical Analysis | url=https://archive.org/details/appliedmultivari00john_130 | url-access=limited | publisher=Prentice Hall | page=[https://archive.org/details/appliedmultivari00john_130/page/n96 76] | isbn=0-13-187715-1 }}</ref>

<math display="block">\operatorname{Var}\left(c^\mathsf{T} X\right) = c^\mathsf{T} \Sigma c .</math>

This implies that the variance of the mean can be written as (with a column vector of ones)

<math display="block">\operatorname{Var}\left(\bar{x}\right) = \operatorname{Var}\left(\frac{1}{n} 1'X\right) = \frac{1}{n^2} 1'\Sigma 1.</math>