在一项对 x 进行 16 次观测的实验中，获得了以下结果： $\sum x = 182$, $\sum x^2 = 3548$ 发现一个观测值 20 记录错误，并已被其正确值 30 替换。求修正后的标准差。

1. Correct the sum of the observations

The incorrect value $20$ is replaced by $30$.

$\sum x = 182 - 20 + 30 = 192$

2. Correct the sum of squares of the observations

$\sum x^2 = 3548 - 20^2 + 30^2$

$= 3548 - 400 + 900 = 4048$

3. Apply the standard deviation formula

There are $n=16$ observations.

$\text{standard deviation}=\sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}$

$=\sqrt{\frac{4048}{16}-\left(\frac{192}{16}\right)^2}$

$=\sqrt{253-12^2}$

$=\sqrt{109}=10.4\text{ correct to 3 significant figures}$