Q90 / 150 · MHT CET 2021 (21 Sep Shift 1) · Mathematics
For two data sets each of size 5 , the variance are 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$\frac{13}{2}$
- B$\frac{5}{2}$
- Correct answerC$\frac{11}{2}$
- D$\frac{15}{2}$
Solution
We have $\mathrm{n}_1=5, \sigma_1{ }^2=4, \overline{\mathrm{x}}_1=2$ and $\mathrm{n}_2=5, \sigma_2{ }^2=5, \overline{\mathrm{x}}_2=4$
here combined mean $=\bar{x}_c=\frac{(5)(2)+(5)(4)}{(5)+(5)}=3$
$$
\mathrm{d}_1=\overline{\mathrm{x}}_1-\overline{\mathrm{x}}_{\mathrm{c}}=2-3=1 \text { and } \mathrm{d}_2=\overline{\mathrm{x}}_2-\overline{\mathrm{x}}_{\mathrm{c}}=4-3=1
$$
Combined variance $=$
$$
=\frac{\mathrm{n}_1\left(\sigma_1^2+\mathrm{d}_1^2\right)+\mathrm{n}_2\left(\sigma_2^2+\mathrm{d}_2^2\right)}{\mathrm{n}_1+\mathrm{n}_2}=\frac{5(4+1)+5(5+1)}{5+5}=\frac{11}{2}
$$
Hence, the correct answer is option: 'C'.