Question

The median of the variables $x+4,x-\frac{7}{2},x-\frac{5}{2}$,
$x-3,x-2,x+\frac{1}{2},x-\frac{1}{2}$ and $x+5$, where $x>0$, is:

• A. $x-3$
• B. $x-2$
• C. $x+\frac{5}{4}$
• D. $x-\frac{5}{4}$

Answer 1

The correct option is D

$x-\frac{5}{4}$

Arrange the given variables in ascending order :
$(x-\frac{7}{2}),(x-3),(x-\frac{5}{2}),(x-2),(x-\frac{1}{2}),(x+\frac{1}{2}),(x+4),(x+5)$
Here, number of observations = 8 [even]
$\therefore \text{Median}=\frac{{\left(\frac{n}{2}\right)}^{th}\text{observation}+{(\frac{n}{2}+1)}^{th}\text{observation}}{2}=\frac{{\frac{8}{2}}^{th}\text{observation}+{(\frac{8}{2}+1)}^{th}\text{observation}}{2}=\frac{4^{th}\text{observation}+5^{th}\text{observation}}{2}=\frac{x-2+x-\frac{1}{2}}{2}=\frac{2x-\frac{5}{2}}{2}=x-\frac{5}{4}$