The following table shows the distribution of marks obtained by a group of $400$ students in an examination.
| Marks less than | $10$ | $20$ | $30$ | $40$ | $50$ | $60$ | $70$ | $80$ | $90$ | $100$ |
|---|---|---|---|---|---|---|---|---|---|---|
| Number of students | $5$ | $10$ | $30$ | $60$ | $105$ | $180$ | $270$ | $355$ | $390$ | $400$ |
Using a scale of $1 cm = 10 $ marks and $1 cm = 50$ students, plot these values and draw a smooth curve through these points.
Estimate from the graph the median marks, the lower quartile marks, the upper quartile marks and the inter-quartile range.
| Marks less than | Class Interval | Number of students (Cumulative frequency) |
|---|---|---|
| $10$ | $0 - 10$ | $5$ |
| $20$ | $10 - 20$ | $10$ |
| $30$ | $20 - 30$ | $30$ |
| $40$ | $30 - 40$ | $60$ |
| $50$ | $40 - 50$ | $105$ |
| $60$ | $50 - 60$ | $180$ |
| $70$ | $60 - 70$ | $270$ |
| $80$ | $70 - 80$ | $355$ |
| $90$ | $80 - 90$ | $390$ |
| $100$ | $90 - 100$ | $400$ |
Number of students $= N = 400$
Taking marks (class intervals) along X-axis and number of students (cumulative frequency) along Y-axis, draw an ogive.
$\therefore \;$ Median marks $= 62.5$
Lower quartile $= Q_1 = \left(\dfrac{N}{4}\right)^{th} \text{value} = \left(\dfrac{400}{4}\right)^{th} \text{value} = 100^{th} \; \text{value} = 49$
$\therefore \;$ Lower quartile $= 49$
Upper quartile $= Q_3 = \left(\dfrac{3N}{4}\right)^{th} \text{value} = \left(\dfrac{3 \times 400}{4}\right)^{th} \text{value} = 300^{th} \; \text{value} = 73.5$
$\therefore \;$ Upper quartile $= 73.5$
Inter-quartile range $= Q_3 - Q_1 = 73.5 - 49 = 24.5$