在一个对 $x$ 进行 16 次观测的实验中，得到以下结果：
$\sum x = 182, \quad \sum x^2 = 3548$
发现一个数值为 20 的观测值被错误记录，并已替换为其正确值 30。求修正后的标准差。 答案 ...

1. Correct the totals for the wrongly recorded value

Replace 20 with 30:
$\sum x = 182 - 20 + 30 = 192$
$\sum x^2 = 3548 - 20^2 + 30^2 = 3548 - 400 + 900 = 4048$

2. Compute the corrected mean and variance

With $n = 16$:
$\text{Mean} = \frac{192}{16} = 12$
$\text{Variance} = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2 = \frac{4048}{16} - 12^2 = 253 - 144 = 109$

3. Find the standard deviation

$\sigma = \sqrt{109} = 10.44030\ldots \approx 10.4$