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

Correct option is A. P = 34, Q = 45

Class interval	Frequency	C.F.
10 - 20	12	12
20 - 30	30	42
30 - 40	$$\displaystyle f_{1}$$	$$\displaystyle 42+f_{1}$$
40 - 50	65	$$\displaystyle 107+f_{1}$$
50 - 60	$$\displaystyle f_{2}$$	$$\displaystyle 107+f_{1}+f_{2}$$
60 - 70	25	$$\displaystyle 132+f_{1}+f_{2}$$
70 - 80	18	$$\displaystyle 150+f_{1}+f_{2}$$

Let the frequency of the class 30 - 40 be $$\displaystyle f_{1}$$ and that of the calss 50 - 60 be $$\displaystyle f_{2}$$ The total frequency is 229
12 + 30 + $$\displaystyle f_{1}$$ + 65 + $$\displaystyle f_{2}$$ + 25 + 18 = 229 $$\displaystyle \Rightarrow f_{1}+f_{2}=79$$
It is given that median is 46 clearly 46 lies in the class 40 - 50. So 40 - 50 is the median class
$$\displaystyle \therefore l=40,h=10,f=65\: and\: C=42+f_{1},N=229$$

Median = $$\displaystyle l+\dfrac{\dfrac{N}{2}-C}{f}\times h\Rightarrow 46=40+\dfrac{\dfrac{229}{2}-(42+f_{1})}{65}\times 10\Rightarrow 46=40+\frac{145-2f_{1}}{13}$$

$$\displaystyle \Rightarrow 6=\dfrac{145-2f_{1}}{13}\rightarrow 2f_{1}=67\Rightarrow f_{1}=33.5\: or\: 34$$ Since $$\displaystyle f_{1}+f_{2}=79\: \: \therefore f_{2}=45$$ Hence $$\displaystyle f_{1}=34$$ and $$\displaystyle f_{2}=45$$