# Quod Erat Demonstrandum

## 2017/09/21

### core math 某題：標準差

Filed under: mathematics,NSS — johnmayhk @ 4:49 下午
Tags:

http://www.wolframalpha.com/input/?i=standard+deviation+of+1,1,1,2,2,2,2,2,3,3,4,4,4,5

http://www.wolframalpha.com/input/?i=standard+deviation+of+8,8,8,9,9,9,9,9,10,10,11,11,11,12

http://www.wolframalpha.com/input/?i=standard+deviation+of+8,9,9,9,10,10,11,11,11,11,11,12,12,12

http://www.wolframalpha.com/input/?i=standard+deviation+of+1,1,1,2,2,2,2,2,3,3,4,4,4,5,5.5

http://www.wolframalpha.com/input/?i=standard+deviation+of+7.5,8,9,9,9,10,10,11,11,11,11,11,12,12,12

（即如何使用數學說服同學？那天，我見部份沒有在堂上睡覺的學生滿臉狐疑，很想多談，但都是走過場算，少在堂上談這些無聊東西，在 blog 談算。）

$\displaystyle \frac{1}{N}((x_1-\overline{x})^2f_1+(x_2-\overline{x})^2f_2 +\dots +(x_n-\overline{x})^2f_n)$

（其中 $N=f_1+f_2+\dots +f_n$

$\displaystyle \frac{1}{N}((-x_1-(-\overline{x}))^2f_1+(-x_2-(-\overline{x}))^2f_2 +\dots +(-x_n-(-\overline{x}))^2f_n)$

$=\displaystyle \frac{1}{N}((\overline{x}-x_1)^2f_1+(\overline{x}-x_2)^2f_2 +\dots +(\overline{x}-x_n)^2f_n)$

$=\displaystyle \frac{1}{N}((x_1-\overline{x})^2f_1+(x_2-\overline{x})^2f_2 +\dots +(x_n-\overline{x})^2f_n)$

$\sigma^2 =\frac{1}{N}\displaystyle \sum_{i=1}^nx_i^2f_i-\overline{x}^2$