Question

Following is the distribution of marks of 50 students in a class:

Marks (more than)	:	0	10	20	30	40	50
No of students	:	50	46	40	20	10	3

Calculate the Median Marks . If 60% of students pass this examination , find out the minimum marks obtained by a pass candidate.

Answer 1

Convert the series into exclusive class interval

C.I.	more than c.f.	f	less than c.f
0 – 10	50	4	4
10 – 20	46	6	10
20 – 30	40	20	30	→ Median class
30 – 40	20	10	40
40 – 50	10	7	47
50 – 60	3	3	50

N = 50
$$\begin{gathered} \mathrm{Median}\mathrm{class}\mathrm{is}\mathrm{given}\mathrm{as}\mathrm{size}\mathrm{of}{\left(\frac{N}{2}\right)}^{\mathrm{th}}\mathrm{item}\mathrm{i}.\mathrm{e}.{25}^{\mathrm{th}}\mathrm{item}\mathrm{which}\mathrm{falls}\mathrm{in}{30}^{\mathrm{th}}\mathrm{cumulative}\mathrm{frequency}\mathrm{group}. \\ \mathrm{Median}=l_1+\frac{\left({\displaystyle \frac{N}{2}}\right)-c.f}{f}\times i \\ =20+\frac{25-10}{20}\times 10 \\ =20+\frac{150}{20} \\ =20+7.5 \\ =27.5 \\ \therefore \boxed{\mathrm{Median}=27.5} \end{gathered}$$

If 60% students pass, it means 40% students fail. Thus, we have to find D₄.
4^th decile class is given by the size of $4{\left(\frac{N}{10}\right)}^{\mathrm{th}}\mathrm{item}$ i.e. 20^th item which falls in 30 cumulative frequency group.
$$\begin{gathered} D_4=20+\frac{20-10}{20}\times 10 \\ =20+\frac{100}{20} \\ =20+5 \\ =25 \\ \therefore \boxed{D_4=25} \end{gathered}$$