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13.Statistics
medium
The average marks of $10$ students in a class was $60$ with a standard deviation $4$, while the average marks of other ten students was $40$ with a standard deviation $6$. If all the $20$ students are taken together, their standard deviation will be
A
$5$
B
$7.5$
C
$9.8$
D
$11.2$
Solution
$\mathrm{n}_{1}=10, \mathrm{n}_{2}=10$
average $\mathrm{m}_{1}=60, \mathrm{m}_{2}=40$
$\sigma_{1}=4, \sigma_{2}=6$
Standard deviation of combined series
$\sigma=\sqrt{\frac{n_{1} \sigma_{1}^{2}+n_{2} \sigma_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}\left(m_{1}-m_{2}\right)^{2}}{\left(n_{1}+n_{2}\right)^{2}}}$
$=\sqrt{\frac{10 \times 16+10 \times 36}{10+10}+\frac{10 \times 10(60-40)^{2}}{(10+10)^{2}}}$
$=\sqrt{8+18+100}=\sqrt{126}=11.2$
Standard 11
Mathematics
