The marks obtained by 100 students in a Mathematics test are given below :

Marks	No. of students
0 - 10	3
10 - 20	7
20 - 30	12
30 - 40	17
40 - 50	23
50 - 60	14
60 - 70	9
70 - 80	6
80 - 90	5
90 - 100	4

Draw an ogive on a graph sheet and from it determine the :

(i) median

(ii) lower quartile

(iii) number of students who obtained more than 85% marks in the test

(iv) number of students who did not pass in the test if the pass percentage was 35.

1. The cumulative frequency table for the given continuous distribution is :

2. Take 1 cm along x-axis = 10 (marks)

3. Take 1 cm along y-axis = 10 (students)

4. Plot the points (10, 3), (20, 10), (30, 22), (40, 39), (50, 62), (60, 76), (70, 85), (80, 91), (90, 96) and (100, 100) representing upper class limits and the respective cumulative frequencies. Also plot the point representing lower limit of the first class i.e. 0 - 10.

5. Join these points by a freehand drawing.

The required ogive is shown in figure above.

(i) Here, n (no. of students) = 100.

To find the median :

Let A be the point on y-axis representing frequency = $\dfrac{n}{2} = \dfrac{100}{2}$ = 50.

Through A draw a horizontal line to meet the ogive at P. Through P, draw a vertical line to meet the x-axis at M. The abscissa of the point M represents 45

Hence, the median marks = 45.

(ii) To find lower quartile :

Let B be the point on y-axis representing frequency = $\dfrac{n}{4} = \dfrac{100}{4}$ = 25.

Through B, draw a horizontal line to meet the ogive at Q. Through Q, draw a vertical line to meet the x-axis at N. The abscissa of the point N represents 32.

Hence, lower quartile = 32.

(iii) Total marks = 100.

85% marks = 85 numbers.

Let O be the point on x-axis representing marks = 85.

Through O draw a vertical line to meet the ogive at R. Through R, draw a horizontal line to meet the y-axis at C. The ordinate of the point C represents 94.

Hence, 94 students score less than 85 so students scoring ,ore than 85 = 100 - 94 = 6.

Hence, 6 students score more than 85% in the test.

(iv) 35% of 100 = 35.

Let T be the point on x-axis representing marks = 35.

Through T, draw a vertical line to meet the ogive at S. Through S, draw a horizontal line to meet the y-axis at D. The ordinate of the point D represents 30.

No. of students who scored less than 35 marks = 30.

Hence, 30 students were failed in the examination.