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)

${\bar{x}}_{old}=10=\frac{\Sigma x_i}{10}\Rightarrow \Sigma x_{\text{i}old}=100$
$\Sigma x_{\text{i}\text{new}}=100-5+15=110$
${\bar{x}}_{\text{new}}=\frac{110}{10}=11$
$Va{r}_{old}=5=\frac{\Sigma x_{iold}^2}{10}-{\left({\left(\bar{x}\right)}_{old}\right)}^2$
$\Sigma x_{iold}^2=1050$
$\Sigma x_{inew}^2=1050-25+225$
$=1250$
${\left(Var\right)}_{new}=\frac{1250}{10}-{\left(11\right)}^2=125-121=4$