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

For $50$ numbers in increasing order, the median is the average of the $25^{\text{th}}$ and $26^{\text{th}}$ terms.Median $M=\frac{25a+26a}{2}=25.5a$.Mean deviation $=\frac{1}{n}\sum |x_i-M|$.
$$50=\frac{1}{50}[|a-25.5a|+|2a-25.5a|+\cdots +|50a-25.5a|]$$
$$2500=|a|[24.5+23.5+\cdots +0.5+0.5+\cdots +24.5]$$
This is $2\times |a|\times [0.5+1.5+\cdots +24.5]$.Sum of AP: $S=\frac{25}{2}(0.5+24.5)=25\times 12.5=312.5$.
$$2500=2\times |a|\times 312.5=625|a|$$
$$|a|=\frac{2500}{625}=4$$