In an experiment with 16 observations on x, the following results were obtained: $\sum x = 182$, $\sum x^2 = 3548$ One observation with value 20 was detected to be wrongly recorded and was replaced by its correct value 30. Find the corrected standard deviation.

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.

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

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

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

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