JEE Advanced

Question 57

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  • #11702
    shahkb4
    Keymaster

    The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2,3,5,10,11,13,15,21, then the mean deviation about the median of all the 10 observations is
    (1) 5
    (2) 4
    (3) 6
    (4) 7

    #11703
    shahkb4
    Keymaster

    Solution

    \[ \begin{aligned} & 2,3,5,10,11,13,15,21, a \& b \\ & \text { Mean }=\frac{\sum x_{i}}{10}=\frac{2+3+5+10+11+13+15+21+a+b}{10}=9 \end{aligned} \]
    \[ \begin{aligned} & \Rightarrow 80+a+b=90 \Rightarrow a+b=10 \\ & \text { Variance }=\frac{1}{n} \sum x_{i}^{2}-(\text { Meau })^{2}=34.2 \\ & \Rightarrow \frac{1}{10}\left(4+9+25+100+121+169+225+441+a^{2}+b^{2}\right)-81=34.2 \\ & \Rightarrow a^{2}+b^{2}=58 \Rightarrow a^{2}+(10-a)^{2}=58 \\ & \Rightarrow 2 a^{2}-20 a+100=58 \Rightarrow a^{2}-10 a+21=0 \\ & \Rightarrow(a-3)(a-7)=0 \\ & a=3 \quad b=7 \\ & b=7 \quad b=3 \end{aligned} \]

    All number 2,3,3,5,7,10,11,13,15,21

    \[ \text { Median }=\frac{\mathrm{T}_{5}+\mathrm{T}_{6}}{2}=\frac{7+10}{2}=8.5 \]

    Mean derivation about median $=\frac{\sum\left|\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right|}{\mathrm{n}}$

    \[ \begin{aligned} & =\frac{6.5+5.5+5.5+3.5+1.5+1.5+2.5+4.5+6.5+12.5}{10} \\ & =5 \text { Answer }(1) \end{aligned} \]
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