Question

The means of two samples of size $40$ and $50$ were found to be $54$ and $63$ respectively. Their standard deviations were $6$ and $9$ respectively. The variance of the combined sample of size $90$ is

• A. $9$
• B. $7$
• C. $90$
• D. $81$

Answer 1

The correct option is D $81$

$\overline{x}=54$
$\Rightarrow \sum x_i=54\times 40=2160$
${\sigma }^2\left(x_i\right)=6^2=36$
$\Rightarrow 36=\frac{\sum x_i^2}{40}-(54{)}^2$
$\Rightarrow \sum x_i^2=118080$

$\overline{y}=63$
$\Rightarrow \sum y_i=63\times 50=3150$
${\sigma }^2\left(y_i\right)=9^2=81$
$\Rightarrow 81=\frac{\sum y_i^2}{50}-(63{)}^2$
$\Rightarrow \sum y_i^2=202500$

$\therefore$ Combined variance,
${\sigma }^2=\frac{1}{40+50}(\sum x_i^2+\sum y_i^2)-{\left(\frac{\sum x_i+\sum y_i}{40+50}\right)}^2$
$=\frac{118080+202500}{90}-{\left(\frac{2160+3150}{90}\right)}^2$
$=3562-3481$
$=81$
$\therefore$ Variance $=81$