Question

If the median of the distribution given below is $28.5$, find the values of $x$ and $y$.

Class interval	Frequency
$0 - 10$	$5$
$10 - 20$	$x$
$20 - 30$	$20$
$30 - 40$	$15$
$40 - 50$	$y$
$50 - 60$	$5$
Total	$60$

Answer 1

Hint: In this question, we are given a table with class interval and frequency. The median of the data is also given. Using the median, find the median class. Draw the table calculating cumulative frequency. Use the median class and formula, Median $ = l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right)h$ to find x. After x has been found, use total frequency from the table to find y.

Formula used: Median $ = l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right)h$ where, $l$ = lower limit of median class, $n = \sum {{f_i}} $, $cf$ = cumulative frequency of the class before median class, $h$ = class interval, $f$ = frequency of median class.

Complete step-by-step solution:
We are given a table with class interval and its frequency and we are asked to find the value of x and y. First, we will make a table with a cumulative frequency (cf).

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

Now, we know that median = $28.5$. Since median lies in the median class, its median class is $20 - 30$. So, we will apply the formula, Median $ = l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right)h$ in this median class.
In this question, $l = 20$, $cf = 5 + x$, $h = 10 - 0 = 10$, $f = 20$ and we know that $n = \sum {{f_i}} = 60$. Therefore, $\dfrac{n}{2} = \dfrac{{60}}{2} = 30$.
Putting all the values in the formula,
$ \Rightarrow 28.5 = 20 + \left( {\dfrac{{30 - (5 + x)}}{{20}}} \right)10$
Solving for x,
$ \Rightarrow 28.5 = 20 + \left( {\dfrac{{30 - 5 - x}}{2}} \right)$
Shifting the terms,
$ \Rightarrow 28.5 - 20 = \left( {\dfrac{{25 - x}}{2}} \right)$
$ \Rightarrow 8.5 \times 2 = 25 - x$
$ \Rightarrow 17 = 25 - x$
Shifting and finding the value of x,
$ \Rightarrow x = 25 - 17 = 8$
Now, we know from the table that $45 + x + y = 60$. Putting $x = 8$ to find the value of y.
$ \Rightarrow 45 + 8 + y = 60$
Shifting to find y,
$ \Rightarrow y = 60 - 53 = 7$

$\therefore $ The value of $x = 8,y = 7$.

Note: Students are often confused while finding cumulative frequency. Cumulative frequency is nothing but the sum of the frequency of that particular interval and the frequencies of predecessors. Basically, it is the sum of all the frequencies before that interval (including the frequency of that interval).