If the median of the following distribution is 58 and sum of all the frequencies is 140. What is the value of x and y?

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Solution

Q.
If the median of the following distribution is 58 and sum of all the frequencies is 140. What is the value of x and y

Class 15 – 25 25 – 35 35 – 45 45 – 55 55 – 65 65 – 75 75 – 85 85 – 95

Frequency 8 10 $x$ 25 40 $y$ 15 7

Ans.
(20)

Variable Frequency c.f.

15-25 8 8

25-35 10 18

35-45 $x$ $18 + x$

45-55 25 $43 + x$

55-65 40 $83 + x$

65-75 $y$ $83 + x + y$

75-85 15 $98 + x + y$

85-95 7 $105 + x + y$

Total $105 + x + y$

And, $105 + x + y = 140$

$\Rightarrow x + y = 35$ ...(i)

Here, Median = 58

Then, median class is 55-65, $l$ = 55, $\frac{N}{2} = \frac{140}{2} = 70$

Then, c.f. = $43 + x$ f = 40

$Median = l + (\frac{\frac{N}{2} - c . f .}{f}) \times h$

$\Rightarrow 58 = 55 + (\frac{70 - 43 - x}{40}) \times 10$

$\Rightarrow 3 = \frac{27 - x}{4} \Rightarrow 12 = 27 - x$

$\Rightarrow x = 27 - 12 = 15 \Rightarrow y = 35 - 15 = 20$

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