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 $4$ , respectively. The variance of the combined data set is :-

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

Answer 1

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

$6x^2=4$ & $6y^2=5$ (given)

$\overline{x}=\frac{\sum x_i}{5}=2$ & $\overline{y}=\frac{\sum y_i}{5}=4$

$6x^2=\left(\frac{\sum x_i^2}{5}\right)=(\overline{x}{)}^2$

$4=\frac{\sum x_i^2}{5}-(2{)}^2$

$8=\frac{\sum x_i^2}{5}$

$\Rightarrow \sum x_i^2=840$

$6y^2=\left(\frac{\sum y_i^2}{5}\right)-(bary{)}^2$

$5=\frac{\sum y{i}^2}{5}-(4{)}^2$

$21\times 5=\sum y{i}^2$

$\Rightarrow \sum y{i}^2=105$

Let variance of combined data is $6z$

$6z^2=\frac{1}{10}(\sum x_i^2+\sum y_i^2)-{\left(\frac{\overline{x}+\overline{y}}{2}\right)}^2$

$=\frac{1}{10}(40+105)-{\left(\frac{2+4}{2}\right)}^2$

$=\frac{145-90}{10}=\frac{55}{10}=\frac{11}{2}$.