If the mean deviation about the median of the numbers $a,2a,3a,\dots ,50a$,is 50, then $\left|a\right|$ equals

Solution:

The numbers are $a,2a,3a,\dots ,50a$. The median of these numbers is $\frac{25a+26a}{2}=255a.$

$\therefore$ Mean deviation about median

$=\frac{\begin{array}{r}|a-25.5a|+|2a-25.5a|+\dots +|-0.5a|+\left|0.5a\right|\\ +\dots +|50a-25.5a|\end{array}}{50}$

$\begin{array}{r}=\frac{\left|a\right|}{25}(0.5+1.5+2.5+\dots +24.5)\\ =\frac{\left|a\right|}{25}\times \frac{25}{2}(0.5+24.5)=\frac{25\left|a\right|}{2}\end{array}$

$\therefore 50=\frac{25\left|a\right|}{2}\Rightarrow \left|a\right|=4$