If the mean deviations about the median of the numbers a, 2a, . . . ,50a is 50, then $\left|\mathrm{a}\right|$ is equal to

Solution:

Median  a, 2a, . . . , 50 is $\frac{25\mathrm{a}+26\mathrm{a}}{2}=(25.5)\mathrm{a}$

Mean deviation about median  $=\frac{\sum _{\mathrm{i}=1}^{50} \mid {\mathrm{x}}_{\mathrm{i}}-\text{ median }\mid }{\mathrm{n}}$

$\begin{array}{l}\Rightarrow 50=\frac{1}{50}\left\{2\right|\mathrm{a}|\cdot (0.5+1.5+2.5+\dots +24.5\left)\right\}\\ \Rightarrow 2500=2\left|\mathrm{a}\right|\frac{25}{2}(2\times 0.5+24\times 1)=2\left|\mathrm{a}\right|\cdot \frac{25}{2}\left(25\right)\\ \Rightarrow \left|\mathrm{a}\right|=4\end{array}$