Question

If the median of the following frequency distribution is 32.5, find the values of f₁ and f₂.

Class interval	0-10	10-20	20-30	30-40	40-50	50-60	60-70	Total
Frequency	f1	5	9	12	f2	3	2	40

Answer 1

Class	Frequency(f)	Cumulative frequency
0-10	$f_1$	$f_1$
10-20	5	$f_1$+5
20-30	9	$f_1$+14
30-40	12	$f_1$+26
40-50	$f_2$	$f_1+f_2$+26
50-60	3	$f_1+f_2$+29
60-70	2	$f_1+f_2$+31
	N=∑f=40

$$\begin{gathered} \begin{array}{l}\text{Now,}{f}_1+f_2+31=40\\ \Rightarrow f_1+f_2=9\\ \Rightarrow f_2=9-f_1...\left(\text{i}\right)\\ \text{The median is 32}\text{.5 which lies in 30-40.}\\ \text{Hence, median class}=30-40\\ \text{Here,}l=30,\frac{N}{2}=\frac{40}{2}=20,f=12\mathrm{and}c.f=14+f_1\\ \text{Now, median}=32.5\\ \Rightarrow l+\frac{\frac{N}{2}-c.f}{f}\times h=32.5\\ \Rightarrow 30+\frac{20-(14+f_1)}{12}\times 10=32.5\end{array} \\ \Rightarrow \frac{6-f_1}{12}\times 10=2.5 \\ \begin{array}{l}\Rightarrow \frac{60-10f_1}{12}=2.5\\ \Rightarrow 60-10f_1=30\\ \Rightarrow 10f_1=30\\ \Rightarrow f_1=3\\ \text{From equation}\left(\text{i}\right),\text{we have:}\\ f_2=9-3\\ \Rightarrow f_2=6\end{array} \end{gathered}$$