Draw an ogive for the data given below and from the graph determine :

(i) the median marks,

(ii) the number of students who obtained more than 75% marks ?

Marks	No. of students
10 - 19	14
20 - 29	16
30 - 39	22
40 - 49	26
50 - 59	18
60 - 69	11
70 - 79	6
80 - 89	4
90 - 99	3

(i) The above distribution is discontinuous, converting into continuous distribution, we get :

Adjustment factor = (Lower limit of one class - Upper limit of previous class) / 2

= $\dfrac{20 - 19}{2} = \dfrac{1}{2}$

= 0.5

Subtract the adjustment factor (0.5) from all the lower limits and add the adjustment factor (0.5) to all the upper limits.

Cumulative frequency distribution table :

Here, n = 120, which is even.

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

= $\dfrac{120}{2}$ = 60th term.

Steps of construction :

1. Since, the scale on x-axis starts at 9.5, a break (kink) is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 9.5

2. Take 1 cm along x-axis = 10 units.

3. Take 1 cm along y-axis = 20 units.

4. Plot the point (9.5, 0) as ogive starts on x-axis representing lower limit of first class.

5. Plot the points (19.5, 14), (29.5, 30), (39.5, 52), (49.5, 78), (59.5, 96), (69.5, 107), (79.5, 113), (89.5, 117) and (99.5, 120).

6. Join the points by a free-hand curve.

7. Draw a line parallel to x-axis from point O (no. of students) = 60, touching the graph at point P. From point P draw a line parallel to y-axis touching x-axis at point Q.

From graph, Q = 42.5 or 43

Hence, median = 42.5 or 43.

(ii) Total marks = 100.

75% marks = $\dfrac{75}{100} \times 100$ = 75.

Draw a line parallel to y-axis from point R (marks) = 75, touching the graph at point S. From point S draw a line parallel to x-axis touching y-axis at point T.

From graph, T = 111.

∴ 111 students score less than or equal to 75%.

Students scoring above than 75% = 120 - 111 = 9.

Hence, 9 students score more than 75%.