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The mean of the numbers $a, b, 8,5,10$ is 6 and their variance is $6.8.$ If $\mathrm{M}$ is the mean deviation of the numbers about the mean, then $25 \mathrm{M}$ is equal to :
Option: 1
$60$

Option: 2
$55$

Option: 3
$50$

Option: 4
$45$

Answers (1)

$\mathrm{\text { mean }=6} \\$

$\mathrm{\frac{a+b+8+5+10}{5}=6 }\\$

$\mathrm{a+b=7} \\$

$\mathrm{b=7-a}$

Variance:

$\mathrm{\sigma^{2}=\frac{\sum_{i=1}^{5}\left(x_{i}-\bar{x}\right)^{2}}{n}}$

$\mathrm{6.8=\frac{(a-6)^{2}+(b-6)^{2}+(8-6)^{2}+(5-6)^{2}+(10-6)^{2}}{5}} \\$

$\mathrm{34=(a-6)^{2}+(7-a-6)^{2}+4+1+18} \\$

$\mathrm{a^{2}-7 a+12=0 \Rightarrow a=4 \text { or } a=3} \\$

$\mathrm{a=4 \quad a=3 }\\$

$\mathrm{b=3 \quad b=4}$

Mean deviation about mean

$\mathrm{M=\frac{\sum_{i=1}^{5}\left|x_{i}-x\right|}{n}} \\$

$\mathrm{M=\frac{|a-6|+|b-6|+|8-6|+|5-6|+|10-6|}{5} } \\$

$\mathrm{\text { when } a=3, b=4 }\\$ and $\mathrm{\text { when } a=4, b=3 } \\$

$\mathrm{M=\frac{3+2+2+1+4}{5}}$ and $\mathrm{\quad M=\frac{2+3+2+1+7}{5}}$

$\mathrm{M=\frac{12}{5}}$ and $\mathrm{M=\frac{12}{5}}$

$\\ \mathrm{25 M=25 \times \frac{12}{5}=60}$

Hece the correct answer is option 1.

Posted by

Sumit Saini

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