Question

Marks obtained by 40 students in an examination are given below:

Marks	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70
No. of students	3	8	14	9	4	2

Using graph paper, draw an ogive and estimate the median marks.

Answer 1

1. Construct the cumulative frequency table

To draw an ogive a cumulative frequency curve, we first need to determine the cumulative frequency (cf) for each class interval based on its upper limit.

Marks (Class interval)	Frequency (f)	Upper Boundary	Cumulative Frequency (cf)
10 – 20	3	20	3
20 – 30	8	30	11 (3 + 8)
30 – 40	14	40	25 (11 + 14)
40 – 50	9	50	34 (25 + 9)
50 – 60	4	60	38 (34 + 4)
60 – 70	2	70	40 (38 + 2)

2. Plot the Ogive

The ogive is plotted by taking the Upper Boundary on the x-axis and the Cumulative Frequency on the y-axis.

1. Coordinate Points: Plot the points (20, 3), (30, 11), (40, 25), (50, 34), (60, 38) and (70, 40).

2. Origin Point: Additionally, plot (10, 0) as the starting point representing the lower limit of the first class.

3. Drawing: Join these points with a smooth, freehand curve to form the “less than” ogive.

3. Estimate the media

The median is the value on the x-axis that corresponds to the middle rank of the data.

1. Find the median rank: Since the total number of students (n) is 40, the median rank is `N/2 = 40/2 = 20`.

2. Locate on graph: Find 20 on the y-axis (Cumulative Frequency). Draw a horizontal line from y = 20 to the ogive curve.

3. Determine x-value: From that point on the curve, draw a vertical line down to the x-axis. The point where this line meets the x-axis is the estimated median marks.

Based on the graph, the horizontal line from 20 on the y-axis meets the curve at a point whose x-coordinate is approximately 36.4.

Thus, the estimated median marks are 36.4.