If you're seeing this message, it means we're having trouble loading external resources on our website.

# 总体与样本标准差回顾

## 总体和样本标准差

• 如果数据本身被视为一个总体, 我们将除以数据点的数量, $N$.
• 如果数据是来自更大数据集的样本, 我们将除以比样本中的数据点少一个的数, $n-1$.

$\sigma =\sqrt{\frac{\sum \left({x}_{i}-\mu {\right)}^{2}}{N}}$

${s}_{x}=\sqrt{\frac{\sum \left({x}_{i}-\overline{x}{\right)}^{2}}{n-1}}$

### 总体标准差

$\sigma =\sqrt{\frac{\sum \left({x}_{i}-\mu {\right)}^{2}}{N}}$

### 例子：总体标准差

$6$, $2$, $3$, $1$

$\mu =\frac{6+2+3+1}{4}=\frac{12}{4}=3$

$6$$6-3=3$
$2$$2-3=-1$
$3$$3-3=0$
$1$$1-3=-2$

$6$$6-3=3$$\left(3{\right)}^{2}=9$
$2$$2-3=-1$$\left(-1{\right)}^{2}=1$
$3$$3-3=0$$\left(0{\right)}^{2}=0$
$1$$1-3=-2$$\left(-2{\right)}^{2}=4$

$9+1+0+4=14$

$\frac{14}{4}=3.5$

$\sqrt{3.5}\approx 1.87$

### 样本标准偏差

${s}_{x}=\sqrt{\frac{\sum \left({x}_{i}-\overline{x}{\right)}^{2}}{n-1}}$

### 例子：样本标准差

$2$, $2$, $5$, $7$

$\overline{x}=\frac{2+2+5+7}{4}=\frac{16}{4}=4$

$2$$2-4=-2$
$2$$2-4=-2$
$5$$5-4=1$
$7$$7-4=3$

$2$$2-4=-2$$\left(-2{\right)}^{2}=4$
$2$$2-4=-2$$\left(-2{\right)}^{2}=4$
$5$$5-4=1$$\left(1{\right)}^{2}=1$
$7$$7-4=3$$\left(3{\right)}^{2}=9$

$4+4+1+9=18$

$\frac{18}{4-1}=\frac{18}{3}=6$

$\sqrt{6}\approx 2.45$