Question

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectivley. 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

$\frac{11}{2}$

${\sigma }_x^2=4,{\sigma }_y^2=5,x=2,y=4\frac{1}{5}\sum x_i^2-(2{)}^2=4;\frac{1}{5}\sum y_i^2-(4{)}^2=5\sum x_i^2=40;\sum y_i^2=105\Rightarrow \sum (x_i^2+y_i^2)=145\Rightarrow \sum (x_i+y_i)=5\left(2\right)+5\left(4\right)=30$

Variance of combined data
$=\frac{1}{10}\sum (x_i^2+y_i^2)-{(\frac{1}{10}\sum (x_i+y_i\left)\right)}^2=\frac{145}{10}-9=\frac{11}{2}$