Normal distribution
Sample mean

Estimator μ ^ {\displaystyle \textstyle {\hat {\mu }}} is called the sample mean, since it is the arithmetic mean of all observations. The statistic x ¯ {\displaystyle \textstyle {\overline {x}}} is complete and sufficient for μ {\textstyle \mu }, and therefore by the Lehmann–Scheffé theorem, μ ^ {\displaystyle \textstyle {\hat {\mu }}} is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally: μ ^ ∼ N (μ, σ 2 / n). {\displaystyle {\hat {\mu }}\sim {\mathcal {N}}(\mu,\sigma ^{2}/n).} The variance of this estimator is equal to the μμ -element of the inverse Fisher information matrix I − 1 {\displaystyle \textstyle {\mathcal {I}}^{-1}}. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of μ ^ {\displaystyle \textstyle {\hat {\mu }}} is proportional to 1 / n {\displaystyle \textstyle 1/{\sqrt {n}}}, 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 simulations.

From the standpoint of the asymptotic theory, μ ^ {\displaystyle \textstyle {\hat {\mu }}} is consistent, that is, it converges in probability to μ {\textstyle \mu } as n → ∞ {\textstyle n\rightarrow \infty }. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples: n (μ ^ − μ) → d N (0, σ 2). {\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu)\,\xrightarrow {d} \,{\mathcal {N}}(0,\sigma ^{2}).}
length_constraints:number_sentences[4], language:response_language[Spanish], oulipo:avoid_letter[e]