The following table shows the marks of $120$ students obtained in an examination. Draw an ogive for the table.
| Marks | $30 - 40$ | $40 - 50$ | $50 - 60$ | $60 - 70$ | $70 - 80$ | $80 - 90$ | $90 - 100$ |
|---|---|---|---|---|---|---|---|
| Number of students | $1$ | $3$ | $11$ | $21$ | $43$ | $32$ | $9$ |
Use the ogive to estimate:
- the median;
- the upper quartile;
- the number of students who got more than $95 \%$ marks;
- the marks obtained by top $20 \%$ students in the examination.
| Marks (Class Interval) | Number of students (frequency $f_i$) | Cumulative Frequency |
|---|---|---|
| $30 - 40$ | $1$ | $1$ |
| $40 - 50$ | $3$ | $4$ |
| $50 - 60$ | $11$ | $15$ |
| $60 - 70$ | $21$ | $36$ |
| $70 - 80$ | $43$ | $79$ |
| $80 - 90$ | $32$ | $111$ |
| $90 - 100$ | $9$ | $120$ |
Number of students $= N = \Sigma f_i = 120$
Taking marks (class intervals) along X-axis and cumulative frequency along Y-axis, draw an ogive.
-
Median $= \left(\dfrac{N}{2}\right)^{th} \text{term} = \left(\dfrac{120}{2}\right)^{th} \text{term} = 60^{th} \text{term} = 75.5$
$\therefore \;$ Median value $= 75.5$ -
Upper quartile $= Q_3 = \left(\dfrac{3N}{4}\right)^{th} \text{term} = \left(\dfrac{3 \times 120}{4}\right)^{th} \text{term} = 90^{th} \text{term} = 83$
$\therefore \;$ Upper quartile $= 83$ -
Number of students who got $95 \%$ marks $= 117$
$\therefore \;$ Number of students who got more than $95 \%$ marks $= 120 - 117 = 3$
-
Number of top $20 \%$ students $= \dfrac{20}{100} \text{ of } 120 = 24$ students
$\therefore \;$ Marks obtained by top $20 \%$ students $= 85 \text{ to } 100$