Question

For two data sets, each of size $5$, the variance is 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. $\frac{5}{2}$
• B. $\frac{11}{2}$
• C. $6$
• D. $\frac{13}{2}$

Answer 1

The correct option is B

$\frac{11}{2}$

Explanation for the correct options:

Step1: Finding the variance

Here, ${{\sigma }_x}^2=4$ and ${{\sigma }_y}^2=5$ ----(1)

$\overline{x}=2$ and $\overline{y}=4$

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

$\sum x_i=10$

Similarly,$\sum _{}^{}{y}_i=20$

${{\sigma }_x}^2=\left[\frac{1}{5}\sum _{}^{}{x_i}^2\right]-4$ ----(2)

${{\sigma }_y}^2=\left[\frac{1}{5}\sum _{}^{}{y_i}^2\right]-16$ ----(3)

From equation(1),(2) and (3), we have

$\Rightarrow$ $\sum {x_i}^2=40$ and $\sum {y_i}^2=105$

Step2:Finding variance of combined data:

Now, Variance of the combined data

$$\begin{gathered} =\frac{1}{10}(40+105)-9 \\ =\frac{145}{10}-9 \end{gathered}$$

$=\frac{55}{10}$

$=\frac{11}{2}$

Hence, Option(B) is correct.