Question

The mean and variance of 10 observation were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ________.

Answer 1

The correct answer is 2
Given,
$\frac{\sum_{i=1}^{10}x_i}{10}= 15\ \ .....(1)$
$⇒$ $\sum_{i=1}^{10} x_i = 150$
and $\frac{\sum_{i=1}^{10} x_{i}^{2}}{10} - 15^2 = 15$
$⇒$ $\sum_{i=1}^{10} x_{2i} = 2400$
Replacing 25 by 15 we get
$⇒$ $\sum_{i=1}^{9} (x_i + 25) = 150$
⇒$$\sum_{i=1}^{9} x_i = 125
∴ Correct mean
= $\frac{\sum_{i=1}^{9}{x_i + 15}}{10} = \frac{125 + 15}{10}$
= 14
Similarly,
$\sum_{i=1}^{2} x_{i}^{2} = 2400 - 25^2$
= 1775
∴ Correct variance = $\frac{\sum_{i=1}^{9} x_{i}^{2} + 15^2}{10} - 14^2$
$= \frac{1775+225}{10}-14^2$
= 4
∴ Correct S.D. $= \sqrt4$
= 2