Question

A sample of $35$ observations has the mean $80$ and S.D. $4$ .A second sample of $65$ observations from the same population has mean $70$ and S.D. $3$ .The S.D.of the combined sample is

• A. $5.85$
• B. $37.2$
• C. $5.58$
• D. None of these

Answer 1

The correct option is A $5.85$

Given $n_1=35,\overline{x_1}=80;{\sigma }_1=4$
$n_2=65,\overline{x_2}=70;{\sigma }_2=3$
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, $\overline{x_{12}}=\frac{n_1{x}_1+n_2{x}_2}{n_1+n_2}$
$\overline{x_{12}}=\frac{35\times 80+65\times 70}{35+65}=73.5$
Substituting the values in the formula for ${\sigma }_{12}$
${\sigma }_{12}=\sqrt{\frac{35(16+{(80-73.5)}^2)+65(9+(70-73.5{)}^2)}{35+65}}$
$\Rightarrow {\sigma }_{12}=5.85$