Question

In the frequency distribution of 100 students gives below. The number of students corresponding to marks groups 10-20 and 30-40 are missing from the table. However, the median is known to be 23. Find the missing frequencies

Marks	0−10	10−20	20−30	30−40	40−50
No. of Students	8	?	40	?	10

Answer 1

Given, Median = 23
N=100
Let the missing frequencies be f₁ and f_2.

Marks	Frequency (f)	Cumulative Frequency (cf)
0 − 10 10 − 20 20 − 30 30 − 40 40 − 50	8 f1 40 f2 10	8 8+f1 48+f1 48+f1+f2 58+f1+f2
	N = ∑f =100

Median class is given by the size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e.${\left(\frac{100}{2}\right)}^{\mathrm{th}}$ item, which is 50^th item.
This corresponds to the class interval of (20 − 30) as median is 23.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{so},23=20+\frac{{\displaystyle \frac{100}{2}}-(8+f_1)}{40}\times 10 \\ \mathrm{or},23=20+\frac{50-8-f_1}{40}\times 10 \\ \mathrm{or},12=42-f_1 \\ \mathrm{or},f_1=42-12 \\ \Rightarrow f_1=30 \end{gathered}$$


58 + f₁ + f₂ = 100
or, f₂ = 100 − 58 − 30
$\Rightarrow$ f₂ = 12

Therefore, f₁ is 30 and f₂ is 12.