Question

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

• A. $\frac{11}{2}$
• B. $6$
• C. $\frac{13}{2}$
• D. $\frac{5}{2}$

Answer 1

The correct option is A $\frac{11}{2}$

Given, $\overline{x}=2$ and $\overline{y}=4$
$\Rightarrow \sum x_i=2\times 5=10$ and $\sum y_i=4\times 5=20$

Also, ${\sigma }_x^2=4$ and ${\sigma }_y^2=5$
$\Rightarrow \frac{1}{5}\sum x_i^2-(2{)}^2=4$ and $\frac{1}{5}\sum y_i^2-(4{)}^2=5$
$\Rightarrow \sum x_i^2=40$ and $\sum y_i^2=105$

Combined variance
$=\frac{1}{5+5}(\sum x_i^2+\sum y_i^2)-{\left(\frac{\sum x_i+\sum y_i}{5+5}\right)}^2$
$=\frac{1}{10}(40+105)-{\left(\frac{10+20}{10}\right)}^2$
$=14.5-9$
$=5.5=\frac{11}{2}$