Editing Variance (section)

====Weighted sum of variables====
{{see also|Weighted arithmetic mean#Variance{{!}}Variance of a weighted arithmetic mean}}
{{distinguish|Weighted variance}}

The scaling property and the Bienaymé formula, along with the property of the [[covariance]] {{math|Cov(''aX'',&nbsp;''bY'') {{=}} ''ab'' Cov(''X'',&nbsp;''Y'')}}  jointly imply that

<math display="block">\operatorname{Var}(aX \pm bY) =a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \pm 2ab\, \operatorname{Cov}(X, Y).</math>

This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if ''X'' and ''Y'' are uncorrelated and the weight of ''X'' is two times the weight of ''Y'', then the weight of the variance of ''X'' will be four times the weight of the variance of ''Y''.

The expression above can be extended to a weighted sum of multiple variables:

<math display="block">\operatorname{Var}\left(\sum_{i}^n a_iX_i\right) = \sum_{i=1}^na_i^2 \operatorname{Var}(X_i) + 2\sum_{1\le i}\sum_{<j\le n}a_ia_j\operatorname{Cov}(X_i,X_j)</math>