Question

The average marks of $10$ students in a class was $60$ with a standard deviation of $4,$ while the average marks of other ten students was $40$ with a standard deviation of $6.$ If all the $20$ students are taken together and $\sigma$ is the combined standard deviation, then the value of $\left[\sigma \right]$ is
($[\cdot ]$ represents the greatest integer function)

Answer 1

$\overline{x}=60$
$\Rightarrow \sum x_i=600$
${\sigma }^2\left(x_i\right)=4^2=16$
$\Rightarrow 16=\frac{\sum x_i^2}{10}-(60{)}^2$
$\Rightarrow \sum x_i^2=36160$

$\overline{y}=40$
$\Rightarrow \sum y_i=400$
${\sigma }^2\left(y_i\right)=6^2=36$
$\Rightarrow 36=\frac{\sum y_i^2}{10}-(40{)}^2$
$\Rightarrow \sum y_i^2=16360$

$\therefore$ Combined variance,
${\sigma }^2=\frac{1}{10+10}(\sum x_i^2+\sum y_i^2)-{\left(\frac{\sum x_i+\sum y_i}{10+10}\right)}^2$
$=\frac{36160+16360}{20}-{\left(\frac{600+400}{20}\right)}^2$
$=2626-2500$
$=126$
$\therefore \left[\sigma \right]=11$