Hi，大家好，我是编程小6，很荣幸遇见你，我把这些年在开发过程中遇到的问题或想法写出来，今天说一说
样本标准差的简单计算公式_样本标准差合理范围是多少,希望能够帮助你!!!。

8个学生分数的样本观察值: 2,4,4,4,5,5,7,9

样本均值:

$$\begin{eqnarray} \overline{x} &=& \frac{\sum x}{N} \nonumber \\ &=& \frac{x_1+x_2+ \cdots +x_n}{N} \nonumber \\ &=& \frac{2+4+4+4+5+5+7+9}{8} \nonumber \\ &=& 5 \nonumber \end{eqnarray}$$



(xi−x¯)2
$(x_i-\overline{x})^2$:

$$\begin{eqnarray} (x_1 - \overline{x})^2 = (2-5)^2 = 9 & (x_2 - \overline{x})^2 = (4-5)^2 =1 \\ (x_3 - \overline{x})^2 = (4-5)^2 = 1 & (x_4 - \overline{x})^2 = (4-5)^2 =1 \\ (x_5 - \overline{x})^2 = (5-5)^2 = 0 & (x_6 - \overline{x})^2 = (5-5)^2 =0 \\ (x_7 - \overline{x})^2 = (7-5)^2 = 4 & (x_8 - \overline{x})^2 = (9-5)^2 =16 \end{eqnarray}$$

方差:

$$\begin{eqnarray} \frac{\sum (x-\overline{x})^2}{N} \nonumber &=& \frac{9+1+1+1+0+0+4+16}{8} \nonumber \\ &=& 4 \nonumber \end{eqnarray}$$

标准差:

$$\sqrt{\frac{\sum{(x-\overline{x})^2}}{N}} = \sqrt{4} = 2$$

上述样本标准差作为总体标准差的估计值偏差较大，称作 Uncorrected sample standard deviation。估计量评选基本标准有 无偏性、有效性、相合性，很难有都满足的情况，一般用 Bessel的纠正方法[2]，作为样本标准差对总体标准差无偏估计量， Corrected sample standard deviation

An unbiased estimator for the variance is given by applying Bessel’s correction, using N − 1 instead of N to yield the unbiased sample variance, denoted $s^2$

$$s^2=\frac{1}{N-1}\sum\limits_{i=1}^N (x-\overline{x})^2$$

标准差:

$$s=\sqrt{\frac{1}{N-1}\sum\limits_{i=1}^N (x-\overline{x})^2}$$

参照：

[1] https://en.wikipedia.org/wiki/Standard_deviation
[2] https://en.wikipedia.org/wiki/Bessel%27s_correction
[3] http://www.techbookreport.com/tutorials/stddev-30-secs.html

今天的分享到此就结束了，感谢您的阅读，如果确实帮到您，您可以动动手指转发给其他人。