Question

The arithmetic mean and standard deviation of a data of nine numbers are $13$ and $5$ respectively. If $3$ is included as the $10$ th item of the data, then the variance of the data of ten number is

• A. 23.5
• B. 21.5
• C. 31.5
• D. 27

Answer 1

Answer: (C)

$\because \frac{\displaystyle \sum_{i=1}^{9} x_{i}}{9}=13$
$\Rightarrow \displaystyle \sum_{i=1}^{9} x_{i}=117.$
and $\sigma^{2}=25=\frac{\displaystyle \sum_{i=1}^{9} x_{i}^{2}}{9}-(13)^{2}$
$\Rightarrow \displaystyle \sum_{i=1}^{y} x_{i}^{2}=9[25+169]$
Now, after including $10^{\text {th }}$ item as $'3'$
New mean $\bar{y}=\frac{\displaystyle \sum_{i=1}^{9} x_{i}+3}{10}=\frac{117+3}{10}=12$
and new variance $=\frac{\left(\displaystyle \sum_{i=1}^{9} x_{i}^{2}+9\right)}{10}-(12)^{2}$
$=\frac{(169+25+1) 9}{10}-144=\frac{1755-1440}{10}=31.5$