From the following cumulative frequency table draw ogive and then use it to find :

(i) Median

(ii) Lower quartile

(iii) Upper quartile

Marks (less than)	Cumulative frequency
10	5
20	24
30	37
40	40
50	42
60	48
70	70
80	77
90	79
100	80

Cumulative frequency distribution table :

Here, n = 80, which is even.

By formula,

Median = $\dfrac{n}{2}$ th term

= $\dfrac{80}{2}$ = 40th term.

Lower quartile = $\dfrac{n}{4}$ th term

= $\dfrac{80}{4}$ = 20th term.

Upper quartile = $\dfrac{3n}{4}$ th term

= $\dfrac{3 \times 80}{4}$ = 60th term.

Steps of construction of ogive :

1. Take 1 cm = 10 units on x-axis.

2. Take 1 cm = 10 units on y-axis.

3. Plot the point (0, 0), as ogive always starts on x-axis representing the lower limit of the first class.

4. Plot the points (10, 5), (20, 24), (30, 37), (40, 40), (50, 42), (60, 48), (70, 70), (80, 77), (90, 79) and (100, 80).

5. Join the points by a free hand curve.

6. Draw a line parallel to x-axis from point L (frequency) = 40, touching the graph at point T. From point T draw a line parallel to y-axis touching x-axis at point M.

7. Draw a line parallel to x-axis from point N (frequency) = 20, touching the graph at point O. From point O draw a line parallel to y-axis touching x-axis at point P.

8. Draw a line parallel to x-axis from point Q (frequency) = 60, touching the graph at point R. From point R draw a line parallel to y-axis touching x-axis at point S.

(i) From graph, M = 40

Hence, median = 40.

(ii) From graph, P = 18

Hence, lower quartile = 18.

(iii) From graph, S = 66

Hence, upper quartile = 66.