Question

A group of 10 observations has mean 5 and S.D. $2\sqrt{6}$ another group of 20 observations has mean 5 and S.D. $3\sqrt{2}$, then the S.D. of combined group of 30 observations is

• A. $\sqrt{5}$
• B. $2\sqrt{5}$
• C. $3\sqrt{5}$
• D. none of these

Answer 1

Answer: (B)

$2\sqrt{5}$

Given $n_1=10,\overline{x_1}=5;{\sigma }_1=2\sqrt{6}$
$n_2=20,\overline{x_1}=5;{\sigma }_2=3\sqrt{2}$
Combined S.D.${\sigma }_{12}=\sqrt{\frac{n_1({{\sigma }_1}^2+{d_1}^2)+n_2({{\sigma }_2}^2+{d_2}^2)}{n_1+n_2}}$
where $d_1=\overline{x_1}-\overline{x_{12}}$ and $d_2=\overline{x_2}-\overline{x_{12}}$
$\overline{x_{12}}$ is combined A.M.
Now, $\overline{x_{12}}=\frac{n_1{x}_1+n_2{x}_2}{n_1+n_2}$
$\overline{x_{12}}=\frac{10\times 5+20\times 5}{10+20}=5$
Substituting the values in the formula for ${\sigma }_{12}$
${\sigma }_{12}=\sqrt{\frac{10(24+{(5-5)}^2)+20(18+(5-5{)}^2)}{10+20}}$
$\Rightarrow {\sigma }_{12}=2\sqrt{5}$