Question

Find the missing frequency form the following distribution, if median is 35 and N = 170.

Variables	0−10	10−20	20−30	30−40	40−50	50−60	60−70
Frequency	10	20	?	40	?	25	15

Answer 1

Given, Median = 35
N=170
Let the missing frequencies be f₁ & f_2.

Class Interval	Frequency (f)	Cumulative Frequency
0 − 10 10 − 20 20 − 30	10 20 f1	10 30 30 + f1 (c.f.)
30 − 40	40 (f)	70 + f1
40 − 50 50 − 60 60 − 70	f2 25 15	70 + f1 + f2 95 + f1 + f2 110 + f1 + f2
	N = ∑f =170

Median class is given by the size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e.${\left(\frac{170}{2}\right)}^{\mathrm{th}}$ item, which is 85^th item.
This corresponds to the class interval of (30 − 40) as median is 35.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{so},35=30+\frac{{\displaystyle \frac{170}{2}}-(30+f_1)}{40}\times 10 \\ \mathrm{or},35=30+\frac{85-30-f_1}{40}\times 10 \\ \mathrm{or},20=55-f_1 \\ \mathrm{or},f_1=35 \\ \Rightarrow f_1=35 \end{gathered}$$


110 + f₁ + f₂ = 170
or, f₂ = 170 − 110 − 35
$\Rightarrow$ f₂ = 25

Therefore, f₁ is 35 and f₂ is 25.