Question

If the median of the following frequency distribution is 46, find the missing frequencies

Variable	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80	Total
Frequency	12	30	P	65	Q	25	18	230

Answer 1

Class interval	Frequency	C.F.
10 - 20	12	12
20 - 30	30	42
30 - 40	$f_1$	$42+f_1$
40 - 50	65	$107+f_1$
50 - 60	$f_2$	$107+f_1+f_2$
60 - 70	25	$132+f_1+f_2$
70 - 80	18	$150+f_1+f_2$

Let the frequency of the class 30 - 40 be $f_1$ and that of the calss 50 - 60 be $f_2$ The total frequency is 229
12 + 30 + $f_1$ + 65 + $f_2$ + 25 + 18 = 229 $\Rightarrow f_1+f_2=79$
It is given that median is 46 clearly 46 lies in the class 40 - 50. So 40 - 50 is the median class
$\therefore l=40,h=10,f=65andC=42+f_1,N=229$

Median = $l+\frac{\frac{N}{2}-C}{f}\times h\Rightarrow 46=40+\frac{\frac{229}{2}-(42+f_1)}{65}\times 10\Rightarrow 46=40+\frac{145-2f_1}{13}$

$\Rightarrow 6=\frac{145-2f_1}{13}\to 2f_1=67\Rightarrow f_1=33.5or34$ Since $f_1+f_2=79\therefore f_2=45$ Hence $f_1=34$ and $f_2=45$