Question

The mean and variance of $10$ observations are found to be $10$ and $4$ respectively. On rechecking it was found that an observation $8$ was incorrect. If it is replaced by $18,$ then the correct variance is

• A. $7$
• B. $8$
• C. $9$
• D. $\frac{55}{6}$

Answer 1

Answer: (C)

$\bar{x}_{(\text {old })}=10=\frac{\Sigma x_{i}}{10} \Rightarrow \Sigma x_{ i (\text { old })}=100$
$\Sigma x_{ i (\text { new })}=100-8+18=110$
$x_{(\text {new })}=\frac{110}{10}=11$
$\operatorname{Var}_{(\text {old })}=4=\frac{\Sigma x_{i(\text { old })}^{2}}{10}-\left(\bar{x}_{(\text {old })}\right)^{2}$
$\Sigma x_{i(\text { old })}^{2}=1040$
$\Sigma x_{i(\text { new })}^{2}=1040-64+324$
$=1300$
$\operatorname{Var}_{(\text {new })}=\frac{1300}{10}-(11)^{2}=130-121=9$