Question

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

• A. $10$
• B. $11$
• C. $12$
• D. $13$

Answer 1

The correct option is B $11$

Given:

${\sigma }_x^2=4$

$\Rightarrow \frac{\sum x_1^2}{n}-{\left(\frac{\sum x_1}{n}\right)}^2=4$
$\Rightarrow \frac{\sum x_1^2}{5}-\left(2\right)2=4$

$\Rightarrow \sum x_1^2=40$

Similarly

$\sum y_1^2=105$
$\therefore {\sigma }^2=\frac{\sum x_i^2+\sum y_i^2}{10}-{\left(\frac{\sum x_i+\sum y_i}{10}\right)}^2$
$=\frac{145}{10}-{\left(\frac{10+20}{10}\right)}^2=5.5$

Double of $5.5$ is $11$.