Editing Normal distribution (section)

==== Sample mean ====
{{See also|Standard error of the mean}}

Estimator <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is called the ''[[sample mean]]'', since it is the arithmetic mean of all observations. The statistic <math style="vertical-align:0">\textstyle\overline{x}</math> is [[complete statistic|complete]] and [[sufficient statistic|sufficient]] for {{tmath|\mu}}, and therefore by the [[Lehmann–Scheffé theorem]], <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is the [[uniformly minimum variance unbiased]] (UMVU) estimator.<ref name="Krishnamoorthy">{{harvtxt |Krishnamoorthy |2006 |p=127 }}</ref> In finite samples it is distributed normally:
<math display=block>
    \hat\mu \sim \mathcal{N}(\mu,\sigma^2/n).
</math>
The variance of this estimator is equal to the ''μμ''-element of the inverse [[Fisher information matrix]] <math style="vertical-align:0">\textstyle\mathcal{I}^{-1}</math>. This implies that the estimator is [[efficient estimator|finite-sample efficient]]. Of practical importance is the fact that the [[standard error]] of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is proportional to <math style="vertical-align:-.3em">\textstyle1/\sqrt{n}</math>, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in [[Monte Carlo simulation]]s.

From the standpoint of the [[asymptotic theory (statistics)|asymptotic theory]], <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is [[consistent estimator|consistent]], that is, it [[converges in probability]] to {{tmath|\mu}} as <math display=inline>n\rightarrow\infty</math>. The estimator is also [[asymptotic normality|asymptotically normal]], which is a simple corollary of the fact that it is normal in finite samples:
<math display=block>
    \sqrt{n}(\hat\mu-\mu) \,\xrightarrow{d}\, \mathcal{N}(0,\sigma^2).
</math>