Question

For two data set each of size $5$, the variances are given to be $4$ and $5$ and the corresponding means are given to be $2$ and $4$ respectively. The variance of the combined data set is

• A. $6$
• B. $13/2$
• C. $5/2$
• D. $11/2$

Answer 1

Answer: (D)

$11/2$

${\sigma }_x^2=4and{\sigma }_y^2=5$
Also $\overline{x}=2and\overline{y}=4$
Now $\frac{\sum x_i}{5}=2\Rightarrow \sum x_i=10$
$\frac{\sum y_i}{5}=4\Rightarrow \sum y_i=20$
Since ${\sigma }_x^2=\frac{1}{5}(\sum x_i^2)-{\left(\overline{x}\right)}^2\Rightarrow \sum x_i^2=40$
Similarly $\sum y_i^2=105$
$\therefore {\sigma }_z^2=\frac{1}{10}(\sum x_i^2+\sum y_i^2)-{\left(\frac{\overline{x}+\overline{y}}{2}\right)}^2$
$=\frac{1}{10}(40+105)-9=\frac{55}{10}=\frac{11}{2}$