Question

The mark of 200 students in a test were recorded as follows:

Marks %	No. of students
10 - 19	7
20 - 29	11
30 - 39	20
40 - 49	46
50 - 59	57
60 - 69	37
70 - 79	15
80 - 89	7

Draw the cumulative frequency table.
Draw an ogive and use it to find:
(i) The median
(ii) The number of students who scored more than 35% marks.

Answer 1

The given frequency distribution is discontinuous, to convert it into continuous distribution.
Adjustment factor = `(20 - 19)/(2)` = 0·5.
Cumulative (continuous) frequency tab;e for the given data is :

Marks % (Classes before adjustment)	Marks % (Classes after adjustment)	Frequency	Cumulative frequency
10 - 19	9·5 - 19·5	7	7
20 - 29	19·5 - 29·5	11	18
30 - 39	29·5 - 39·5	20	38
40 - 49	39·5 - 49·5	46	84
50 - 59	49·5 - 59·5	57	141
60 - 69	59·5 - 69·5	37	178
70 - 79	69·5 - 79·5	15	193
80 - 89	79·5 - 89·5	7	200

Take 1 cm along X-axis = 10% marks and 1 cm along Y-axis = 25 students.
Plot the point (19·5, 7), (29·5 - 18), (39·5 - 38), (49·5 - 141), (69·5 - 178), (9·5 - 193), (89·5 - 200) and (9·5 - 0) join these points by a free hand drawing.
The required ogive is drawn in the figure given below:

(i) To find the median: Let A be a point on Y-axis representing frequency
= `(1)/(2) [("n"^"th"/2 "term") + ("n"/2 + 1)^"th" "term"]`
= `(1)/(2)(100 + 101)`
= 100·5.
Through A draw a horizontal line to meet the ogive at P. Through P draw a vertical line to meet X-axis at M. the abscissae of point M represents 52%.
∴ The required median = 52%.
(ii) Let the point B on X-axis represent 35% marks. Through B draw a vertical line to meet the ogive at Q. Through Q draw a horizontal line to meet Y-axis at C. The ordinate of the point C represents 28 students on Y-axis.
∴ The number of students who scored more than 35% marks = total no. of students - no. of students who scored ≤35%
= 200 - 8
= 172.