JEE Main 2022 — Statistics Question with Solution

Median is the central value (or middle observation) of statistical data if it is arranged in ascending or descending order.

Thus, if there are $n$ observations (variates) $x_1, x_2, x_3, \dots , x_n$ arranged in ascending or descending order, then

median $={\left(\frac{n+1}{2}\right)}^{\mathrm{th}}$ observation, if $n$ is odd

and median $=\frac{\frac{n}{2}\text{ th observation }+\left(\frac{n}{2}+1\right)\text{ th observation }}{2}$, if $n$ is even.

Here we can see there are even number of terms,

So, median $=\frac{2k+12}{2}=k+6$

Mean deviation $=\sum \frac{\left|x_i-M\right|}{n}=6$

$$$\Rightarrow \frac{\left(k+3\right)+\left(k+1\right)+\left(k-1\right)+\left(6-k\right)+\left(6-k\right)+\left(10-k\right)+\left(15-k\right)+\left(18-k\right)}{8}=6$

$\Rightarrow \frac{58-2k}{8}=6$

$\Rightarrow k=5$

So, median $=\frac{2\times 5+12}{2}=11$