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

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Given:

The median of the distribution given is 28.5. The total frequency is 60.

To do:

We have to find the values of $x$ and $y$.

Solution:

Median $= 28.5$ and $N = 60$

$x + y = 60-(5+20+15+5)$

$x+y = 60 - 45 = 15$

$y = 15-x$.....….(i)

The cumulative frequency (CF) table for the following data is as below:

Here,

Number of observations $(n) = 60$

So,

$\frac{n}{2}$ ​= 30

Since the median is 28.5, the median class is 20−30

Hence, $l = 20, h = 10, f = 20, cf = 5 + x$.

Therefore,

Median $= l+(\frac{\frac{n}{2} \ -\ cf}{f}) h$

$28.5 = 20\ + (\frac{30\ \ -\ \ 5\ \ -\ \ x}{20})\times \ 10$

$28.5 = 20\ + (\frac{25\ \ -\ \ x}{2})$

$28.5\ -\ 20 = (\frac{25\ \ -\ \ x}{2})$

$8.5\times2 = 25\ −\ x$

$x = 8$  

This implies,

$y=15-8$               [From (i)]

$y = 7$

The values of $x$ and $y$ are $8$ and $7$ respectively.