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

Correct Answer: 2

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