Question

The mean and variance of $10$ observations are found to be $10$ and $5$ respectively. On rechecking it is found that an observation $5$ is incorrect. If the incorrect observation is replaced by $15,$ then the correct variance is

• A. $7$
• B. $8$
• C. $9$
• D. $4$

Answer 1

Answer: (D)

$\bar{x}_{o l d}=10=\frac{\Sigma x_{i}}{10}\Rightarrow \Sigma x_{\textit{i} o l d}=100$
$\Sigma x_{\textit{i} \text{new}}=100-5+15=110$
$\bar{x}_{\text{new}}=\frac{110}{10}=11$
$Var_{old}=5=\frac{\Sigma x_{i o l d}^{2}}{10}-\left(\left(\bar{x}\right)_{o l d}\right)^{2}$
$\Sigma x_{i o l d}^{2}=1050$
$\Sigma x_{i new}^{2}=1050-25+225$
$=1250$
$\left(Var\right)_{n e w}=\frac{1250}{10}-\left(11\right)^{2}=125-121=4$