Question

If the median of a distribution given below is $28.5$ then, find the value of $\mathrm{x}$ & $\mathrm{y}$.

Class Interval	Frequency
$0-10$	$5$
$10-20$	$\mathrm{x}$
$20-30$	$20$
$30-40$	$15$
$40-50$	$\mathrm{y}$
$50-60$	$5$
Total	$60$

Answer 1

Step 1: Create a cumulative frequency table

Class Interval	Frequency	Cumulative Frequency
$0-10$	$5$	$5$
$10-20$	$\mathrm{x}$	$5+\mathrm{x}$
$20-30$	$20$	$25+\mathrm{x}$
$30-40$	$15$	$40+\mathrm{x}$
$40-50$	$\mathrm{y}$	$40+\mathrm{x}+\mathrm{y}$
$50-60$	$5$	$45+\mathrm{x}+\mathrm{y}$
Total	$60$

Step 2: Find the value of $\mathrm{x}$

Median = $28.5$

$\mathrm{n}=60$

Median lies in a class interval $20-30$.

Cumulative frequency ${\mathrm{C}}_{\mathrm{f}}=25+\mathrm{x}$

$$\begin{gathered} \mathrm{Lower}\mathrm{limit}\mathrm{l}=20 \\ \mathrm{Higher}\mathrm{limit}\mathrm{r}=30 \\ \mathrm{Class}\mathrm{size}\mathrm{h}=10 \\ \mathrm{Frequency}=20 \end{gathered}$$

That is

$$\begin{gathered} 28.5=20+\frac{30-25+x}{20}\times 10[\mathrm{Median}=\mathrm{l}+\frac{{\displaystyle \frac{\mathrm{n}}{2}-{\mathrm{C}}_{\mathrm{f}}}}{\mathrm{f}}\times \mathrm{h}] \\ 28.5=20+\frac{5+x}{2} \\ 28.5=\frac{40+5+x}{2} \\ 28.5\times 2=45+x \\ 57-45=x \\ 12=x \end{gathered}$$

Step 3: Find the value of $\mathrm{y}$

The sum of cumulative frequency is $45+\mathrm{x}+\mathrm{y}$, we can write that

$$\begin{gathered} 45+\mathrm{x}+\mathrm{y}=60 \\ \mathrm{y}=60-45-\mathrm{x} \\ \mathrm{y}=60-45-12 \\ \mathrm{y}=3 \end{gathered}$$

Hence, the value of $\mathrm{x}$ and $\mathrm{y}$ is $12,3$.