MHT CET · Maths · Statistics
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 \(\frac{5}{2}\)
- C \(\frac{11}{2}\)
- D \(\frac{15}{2}\)
Answer & Solution
Correct Answer
(C) \(\frac{11}{2}\)
Step-by-step Solution
Detailed explanation
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 \(=\)
\(
=\frac{\mathrm{n}_1\left(\sigma_1^2+\mathrm{d}_1^2\right)+\mathrm{n}_2\left(\sigma_2^2+\mathrm{d}_2^2\right)}{\mathrm{n}_1+\mathrm{n}_2}=\frac{5(4+1)+5(5+1)}{5+5}=\frac{11}{2}
\)
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 \(=\)
\(
=\frac{\mathrm{n}_1\left(\sigma_1^2+\mathrm{d}_1^2\right)+\mathrm{n}_2\left(\sigma_2^2+\mathrm{d}_2^2\right)}{\mathrm{n}_1+\mathrm{n}_2}=\frac{5(4+1)+5(5+1)}{5+5}=\frac{11}{2}
\)
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