样本标准差的简单计算

8个学生分数的样本观察值: 2,4,4,4,5,5,7,9

样本均值:

x¯====∑xNx1+x2+⋯+xnN2+4+4+4+5+5+7+985

(xi−x¯)2 :

(x1−x¯)2=(2−5)2=9(x3−x¯)2=(4−5)2=1(x5−x¯)2=(5−5)2=0(x7−x¯)2=(7−5)2=4(x2−x¯)2=(4−5)2=1(x4−x¯)2=(4−5)2=1(x6−x¯)2=(5−5)2=0(x8−x¯)2=(9−5)2=16

方差:

∑(x−x¯)2N==9+1+1+1+0+0+4+1684

标准差:

∑(x−x¯)2N−−−−−−−−−−√=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 s2

s2=1N−1∑i=1N(x−x¯)2

标准差:

s=1N−1∑i=1N(x−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