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

Answer 1

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

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