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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 |
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:
CI | f | CF |
0-10 | $5$ | $5$ |
10-20 | $x$ | $5\ +\ x$ |
20-30 | $20$ | $25\ +\ x$ |
30-40 | $15$ | $40\ +\ x$ |
40-50 | $y$ | $40\ +\ x\ +\ y$ |
50-60 | $5$ | $45\ +\ x\ +\ y$ |
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.
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