Question

Median of an even-numbered non-negative data set is $3.$ Determine the number of sets of values for the two central values of the data set when the data set is arranged in ascending order.

• A. Zero
• B. Three
• C. Four
• D. Seven

Answer 1

The correct option is C Four

Consider a even number data set $\{a,b,c,d\}$ such that $a\le b\le c\le d.$

Median = Average of second and third data points
$\Rightarrow$ Median $=\frac{b+c}{2}$
$\Rightarrow 3=\frac{b+c}{2}$
$\Rightarrow b+c=6$

We know, $6=0+6$
$6=1+5$
$6=2+4$
$6=3+3$
$6=4+2$ and so on.

However, $c$ cannot be less than $b$, i.e., $3.$
$\therefore b+c=6=0+6\Rightarrow b=0,c=6$
or, $b+c=6=1+5\Rightarrow b=1,c=5$
or, $b+c=6=2+4\Rightarrow b=2,c=4$
or, $b+c=6=3+3\Rightarrow b=3,c=3$

Hence, the possible pairs of two central values are $\{0,6\},\{1,5\},\{2,4\}$ and $\{3,3\}$.

$\therefore$ There are four possible pairs.