Editing Variance (section)

====Sum of uncorrelated variables====
{{main article|Bienaymé's identity}}
{{see also|Sum of normally distributed random variables}}
One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of [[uncorrelated]] random variables is the sum of their variances:

<math display="block">\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i).</math>

This statement is called the [[Irénée-Jules Bienaymé|Bienaymé]] formula<ref>[[Michel Loève|Loève, M.]] (1977) "Probability Theory", ''Graduate Texts in Mathematics'', Volume 45, 4th edition, Springer-Verlag, p.&nbsp;12.</ref> and was discovered in 1853.<ref>[[Irénée-Jules Bienaymé|Bienaymé, I.-J.]] (1853) "Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés", ''Comptes rendus de l'Académie des sciences Paris'', 37, p.&nbsp;309–317; digital copy available [http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-2994&I=313] {{Webarchive|url=https://web.archive.org/web/20180623145935/http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-2994&I=313|date=2018-06-23}}</ref><ref>[[Irénée-Jules Bienaymé|Bienaymé, I.-J.]] (1867) "Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés", ''Journal de Mathématiques Pures et Appliquées, Série 2'', Tome 12, p.&nbsp;158–167; digital copy available [http://gallica.bnf.fr/ark:/12148/bpt6k16411c/f166.image.n19][http://sites.mathdoc.fr/JMPA/PDF/JMPA_1867_2_12_A10_0.pdf]</ref> It is often made with the stronger condition that the variables are [[statistical independence|independent]], but being uncorrelated suffices. So if all the variables have the same variance σ<sup>2</sup>, then, since division by ''n'' is a linear transformation, this formula immediately implies that the variance of their mean is

<math display="block">
  \operatorname{Var}\left(\overline{X}\right) = \operatorname{Var}\left(\frac{1}{n} \sum_{i=1}^n X_i\right) =
  \frac{1}{n^2}\sum_{i=1}^n \operatorname{Var}\left(X_i\right) =
  \frac{1}{n^2}n\sigma^2 =
  \frac{\sigma^2}{n}.
</math>

That is, the variance of the mean decreases when ''n'' increases. This formula for the variance of the mean is used in the definition of the [[standard error (statistics)|standard error]] of the sample mean, which is used in the [[central limit theorem]].

To prove the initial statement, it suffices to show that

<math display="block">\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y).</math>

The general result then follows by induction.  Starting with the definition,

<math display="block">\begin{align}
  \operatorname{Var}(X + Y) &= \operatorname{E}\left[(X + Y)^2\right] - (\operatorname{E}[X + Y])^2 \\[5pt]
    &= \operatorname{E}\left[X^2 + 2XY + Y^2\right] - (\operatorname{E}[X] + \operatorname{E}[Y])^2.
\end{align}</math>

Using the linearity of the [[Expectation Operator|expectation operator]] and the assumption of independence (or uncorrelatedness) of ''X'' and ''Y'', this further simplifies as follows:

<math display="block">\begin{align}
   \operatorname{Var}(X + Y) &= \operatorname{E}{\left[X^2\right]} + 2\operatorname{E}[XY] + \operatorname{E}{\left[Y^2\right]} - \left(\operatorname{E}[X]^2 + 2\operatorname{E}[X] \operatorname{E}[Y] + \operatorname{E}[Y]^2\right) \\[5pt]
   &= \operatorname{E}\left[X^2\right] + \operatorname{E}\left[Y^2\right] - \operatorname{E}[X]^2 - \operatorname{E}[Y]^2 \\[5pt]
   &= \operatorname{Var}(X) + \operatorname{Var}(Y).
\end{align}</math>