Question

If the median of the following frequency distribution is 32.5, find the values of $f_1$ and $f_2$ $\begin{array}{ccccccccc}\text{Class}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& \text{Total}\\ \text{interval}& & & & & & & & \\ \text{Frequency}& f_1& 5& 9& 12& f_2& 3& 2& 40\end{array}$

Answer 1

We prepare the cumulative frequency table, as shown below:

Class	Frequency	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=Σf=40

Now, $f_1+f_2+31=40$

⇒$f_1+f_2=9$

⇒$f_2=9–f_1$.......(i)

The median is 32.5 which lies in 30 – 40.

Hence, median class = 30 – 40

Therefore, $l=30,N/2=20,f=12,andcf=14+f_1$

Now,

$Median=l+\left(\right(N/2-cf\left)/f\right)\times h$

$32.5=30+\left(\right(20–(14+f_1)\left)/12\right)\times 10$

⇒$32.5=30+\left(\right(6–f_1\left)/12\right)\times 10$

⇒$2.5=\left(\right(6–f_1\left)/12\right)\times 10$

⇒$60–10f_1=30$

⇒$10f_1=30$

⇒$f_1=3$

From eq. (i), we have

$f_2=9-3$
⇒$f_2=6$