Question

Two groups of $50$ students were formed. It was observed that Group I had mean and SD $60and8$ respectively, that of group II was $55and7$. Find mean and SD of $100$ students.

• A. $57.5,7.92$
• B. $57,8$
• C. $56.5,7$
• D. $55.7,9.72$

Answer 1

The correct option is A $57.5,7.92$

$n_1=50,{\overline{x}}_1=60,{\sigma }_1=8,$ $n_2=50,{\overline{x}}_2=60,{\sigma }_2=8$

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}=\frac{50\times 60+50\times 55}{100}=57.5$

$d_1={\overline{x}}_1-\overline{x}=60-57.5=2.5$

$d_2={\overline{x}}_2-\overline{x}=55-57.5=-2.5$

${\sigma }^2(n_1+n_2)=n_1({\sigma }_1^2+d_1^2)+n_2({\sigma }_2^2+d_2^2)$

${\sigma }^2\times \left(100\right)=50\times (64+6.25)+50\times (49+6.25)$

${\sigma }^2=\frac{3512.5+2762.5}{100}=62.75\sigma =7.92$