贝塞尔无偏估计校正

The sample mean is given by
\begin{align}
\overline{x}=\frac{1}{n}\sum_{i=1}^n x_i.
\end{align}

The biased sample variance is then written:
\begin{align}
s_n^2 = \frac {1}{n} \sum_{i=1}^n \left(x_i - \overline{x} \right)^ 2 = \frac{\sum_{i=1}^n \left(x_i^2\right)}{n} - \frac{\left(\sum_{i=1}^n x_i\right)^2}{n2}
\end{align}

and the unbiased sample variance is written:
\begin{align}
s^2 = \frac {1}{n-1} \sum_{i=1}^n \left(x_i - \overline{x} \right)^ 2 = \frac{\sum_{i=1}^n \left(x_i^2\right)}{n-1} - \frac{\left(\sum_{i=1}^n x_i\right)^2}{(n-1)n} = \left(\frac{n}{n-1}\right),s_n^2.
\end{align}

Recycling an identity for variance,
\begin{align}
\sum_{i=1}^{n}\left( x_{i}-\overline{x}\right)^{2} & = \sum_{i=1}^{n} \left(x_{i}^{2}-2x_{i} \overline{x} + \overline{x}^2 \right) \nonumber \\
& = \sum_{i=1}^{n} x_{i}^{2}-2 \overline{x} \sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} \overline{x}^{2} \nonumber \\
&=\sum_{i=1}^{n}x_{i}-2n \overline{x}^{2} + n{\overline {x}}^{2} \nonumber \\
&=\sum_{i=1}^{n}x_{i}-n\overline{x}^{2}
\end{align}

so

\begin{align}
\operatorname{E} \left(\sum_{i=1}^{n}\left[(x_{i}-\mu )-\left({\overline {x}}-\mu \right)\right]^{2}\right) & = \operatorname{E} \left(\sum_{i=1}^{n}(x_{i}-\mu )^{2}-n({\overline{x}}-\mu )^{2}\right) \nonumber \\
& = \sum_{i=1}^{n} \operatorname{E} \left((x_{i}-\mu )^{2}\right)-n \operatorname{E} \left(({\overline {x}}-\mu )^{2}\right) \nonumber \\
& = \sum_{i=1}^{n} \operatorname{Var} \left(x_{i}\right) - n \operatorname{Var} \left({\overline {x}} \right)
\end{align}

and by definition,

\begin{align}
\operatorname{E} (s^{2}) & = \operatorname{E} \left(\sum_{i=1}^{{n}{\frac{(x_{i}-{\overline{x}})}{2}}{n-1}}\right) \nonumber \\
& = {\frac{1}{n-1}}\operatorname{E} \left(\sum_{i=1}^{n}\left[(x_{i}-\mu )-\left({\overline{x}}-\mu \right)\right]^{2}\right) \nonumber \\
& = {\frac{1}{n-1}} \left[\sum_{i=1}^{n} \operatorname{Var} \left(x_{i}\right) - n\operatorname{Var} \left({\overline{x}} \right)\right]
\end{align}

Reference：https://en.wikipedia.org/wiki/Bessel's_correction

关于cnblog的markdown公式写法，插入两条reference
https://www.zybuluo.com/byjr-k/note/253737
http://code-kid.com/2014/04/06/2014-04-06-用markdown写公式/
http://m.it610.com/article/4852527.htm