Question

If the mean deviation about the median of the numbers $a,2a,....,50a$ is $50$. Then $|a|$ equals

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is C $4$

$a,2a,....,50a$ are the observation so, total no. of observation is $50$
So, median $M=\frac{{\left(\frac{50}{2}\right)}^{th}+{(\frac{50}{2}+1)}^{th}}{2}=\frac{25a+26a}{2}=\frac{51}{2}$
Mean deviation about median=$\frac{\sum _{i=1}^n|x_i-M|}{n},n=50$
Put the value of $x_i$ and $M$
So, $50=\frac{|a-25.5a|+|2a-25.5a|+...+|50a-25.5a|}{50}\Rightarrow 50=\frac{24.5|a|+23.5|a|+...0.5|a|+0.5|a|+....+24.5|a|}{50}$
$\Rightarrow 50=\frac{2(0.5|a|+1.5|a|+...+24.5|a|}{50}\Rightarrow 50=\frac{|a|(1+2+...+49}{50}=\frac{|a|\frac{25}{2}(1+49)}{50}|a|=\frac{50\times 50}{25\times 25}=4$