Calculate the missing frequency from the following distribution, … - Vidyadip

Question

Calculate the missing frequency from the following distribution, it being given that the median of the distribution is $24.$

Class	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequency	$5$	$25$	$?$	$18$	$7$

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Answer

Class	Frequency (f)	Cumulative Frequency
$0-10$	$5$	$5$
$10-20$	$25$	$30$
$20-30$	$x$	$x + 30$
$30-40$	$18$	$x + 48$
$40-50$	$7$	$x + 55$

Median is $24$ which lies in $20-30$
$\therefore$ Median Class $= 20-30$
Here, $\text{l}=20,\ \frac{\text{n}}{2}=\frac{\text{x}+55}{2},$ c. f of the preceding class $\text{c.f}=30,\ \text{f}=\text{x},\ \text{h}=10$
$\therefore$ Median $\text{l}+\frac{\frac{\text{n}}{2}-\text{c.f}}{\text{f}}\times\text{h}$
$\Rightarrow24=20+\frac{\frac{\text{x}+55}{2}-30}{\text{x}}\times\text{10}$
$\Rightarrow24=20+\frac{\frac{\text{x}+55-60}{2}}{\text{x}}\times10$
$\Rightarrow24=20+\frac{\text{x}-5}{2\text{x}}\times10$
$\Rightarrow24=20+\frac{5\text{x}-25}{\text{x}}$
$\Rightarrow24=\frac{20\text{x}+5\text{x}-25}{\text{x}}$
$\Rightarrow24\text{x}=25\text{x}=25$
$\Rightarrow-\text{x}=-25$
$\Rightarrow\text{x}=25$
Hence, the unknown frequency is $25$

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