The marks obtained by $80$ students in a test are given below:
| Marks | $0 - 10$ | $10 - 20$ | $20 - 30$ | $30 - 40$ | $40 - 50$ | $50 - 60$ | $60 - 70$ | $70 - 80$ |
|---|---|---|---|---|---|---|---|---|
| Number of students | $3$ | $7$ | $15$ | $24$ | $16$ | $8$ | $5$ | $2$ |
Draw an ogive for the given distribution on a graph paper. Use a scale of $1 \; cm = 10$ units on both the axes. Estimate from the ogive
- the median value;
- the lower quartile value;
- the number of students who obtained more than $65$ marks;
- the number of students who did not pass in the test if the pass percentage marks was $35 \%$ of the maximum marks.
| Marks (Class Interval) | Number of students (frequency) | Cumulative frequency |
|---|---|---|
| $0 - 10$ | $3$ | $3$ |
| $10 - 20$ | $7$ | $10$ |
| $20 - 30$ | $15$ | $25$ |
| $30 - 40$ | $24$ | $49$ |
| $40 - 50$ | $16$ | $65$ |
| $50 - 60$ | $8$ | $73$ |
| $60 - 70$ | $5$ | $78$ |
| $70 - 80$ | $2$ | $80$ |
Number of students $= N = 80$
- Median value $= \left(\dfrac{N}{2}\right)^{th}$ value $= \left(\dfrac{80}{2}\right)^{th}$ value $= 40^{th}$ value $= 36$
$\therefore \;$ Median value $= 36$ - Lower quartile $= \left(\dfrac{N}{4}\right)^{th}$ value $= \left(\dfrac{80}{4}\right)^{th}$ value $= 20^{th}$ value $= 27$
$\therefore \;$ Lower quartile $= 27$ - From the ogive, number of students who got $65$ marks $= 76$
$\therefore \;$ Number of students who got more than $65$ marks $= 80 - 76 = 4$ students - Pass marks $= 35 \% \text{ of } 80 = \dfrac{35}{100} \times 80 = 28$ marks
$\therefore \;$ From the ogive, number of students who did not pass $= 21$ students