The marks obtained by 120 students in a Mathematics test are given below :

Marks	No. of students
0 - 10	5
10 - 20	9
20 - 30	16
30 - 40	22
40 - 50	26
50 - 60	18
60 - 70	11
70 - 80	6
80 - 90	4
90 - 100	3

Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive. Use your ogive to estimate :

(i) the median

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

(iii) the number of students who did not pass in the test if the pass percentage was 40?

(iv) the lower quartile.

Cumulative frequency distribution table :

(i) Steps of construction of ogive :

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

2. Take 1 cm = 20 students on y-axis.

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

4. Plot the points (10, 5), (20, 14), (30, 30), (40, 52), (50, 78), (60, 96), (70, 107), (80, 113), (90, 117) and (100, 120).

5. Join the points by a free hand curve.

6. Draw a line parallel to x-axis from point A (no. of students) = 60, touching the graph at point B. From point B draw a line parallel to y-axis touching x-axis at point C.

From graph, C = 43

Hence, median = 43.

(ii) Total marks = 100.

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

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

From graph, F = 110.

It means that 110 students score either less or equal to 75% marks.

No. of students left = 120 - 110 = 10.

Hence, no. of students scoring more than 75% marks = 10.

(iii) Total marks = 100.

40% of 100 marks = $\dfrac{40}{100} \times 100$ = 40.

Draw a line parallel to y-axis from point G (marks) = 40, touching the graph at point H. From point H draw a line parallel to x-axis touching y-axis at point I.

From graph, I = 52.

Hence, no. of failed students = 52.

(iv) Here, n = 120, which is even.

By formula,

Lower quartile = $\dfrac{n}{4} = \dfrac{120}{4}$ = 30th term.

Draw a line parallel to x-axis from point J (no. of students) = 30, touching the graph at point K. From point K draw a line parallel to y-axis touching x-axis at point L.

From graph, L = 30

Hence, lower quartile = 30.