## Ogive (Cumulative frequency curve)

Ogive is a graphic presentation of the cumulative frequency distribution of continuous series. In a continuous series, if the upper limit (or the lower limit) of each class interval is taken as x-coordinate and its corresponding c.f. as y-coordinate and the points are plotted in the graph, we obtain a curve by joining the points with freehand. Such a curve is known as ogive or cumulative frequency curve.

Since ‘less than’ c.f. and ‘more than’ c.f. are the two types of cumulative frequency distribution, there are two types of ogive. They are less than ogive and more than ogive.

1.   Less than ogive (or less than cumulative frequency curve)

When the upper limit of each class interval is taken as x-coordinate and its corresponding frequency as y-coordinate, the ogive so obtained is known as less than ogive (or less than cumulative frequency curve). Obviously, less than ogive is an increasing curve, sloping upwards from left to right and has the shape of an elongated S.

2.   More than ogive (or more than cumulative frequency curve)

When the lower limit of each class interval is taken as x-coordinate and its corresponding frequency as y-coordinate, the ogive so obtained is known as more than ogive (or more than cumulative frequency curve). More than ogive is a decreasing curve sloping downward from left to right and has the shape of an elongated S, upside down.

### Construction of less than ogive and more than ogive:

To construct a ‘less than’ or ‘more than’ ogive, we proceed as follows:

1)    Make a less than cumulative frequency table.

2)    Choose the suitable scale and make the upper class limit of each class interval along x-axis and cumulative frequencies along y-axis.

3)    Plot the coordinates (upper limit, less than c.f.) on the graph.

4)    Join the point by freehand and obtain a less than ogive.

In the case of more than ogive, we should prepare the more than cumulative frequency table. The lower class limits of each class interval are marked on x-axis. Then the process is similar to the construction of less than ogive.

### Worked Out Examples

Example 1: The table given below shows the marks obtained by 80 students in science. Construct (i) less than ogive  (ii) more than ogive.

 Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of students 3 8 17 29 15 6 2

Solution: Here,

Less than cumulative frequency table:

 Marks No. of students Upper limit Less than c.f. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 3 8 17 29 15 6 2 10 20 30 40 50 60 70 3 (less than 10) 11 (less than 20) 28 (less than 30) 57 (less than 40) 72 (less than 50) 78 (less than 60) 80 (less than 70)

The less than ogive graph,

More than cumulative frequency table:

 Marks No. of students Upper limit More than c.f. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 3 8 17 29 15 6 2 0 10 20 30 40 50 60 80 (more than 0) 80-3 = 77 (more than 10) 77-8 = 69 (more than 20) 69-17 = 52 (more than 30) 52-29 = 23 (more than 40) 23-15 = 8 (more than 50) 8-6 = 2 (more than 60)

The more than ogive graph,

If we draw the both less than ogive and more than ogive of a distribution on the same graph paper, they intersect at a point. The foot of the perpendicular drawn from the point of intersection of two ogives to the x-axis gives the value of median of the distribution.

For example,

 Marks f Upper limit Less than c.f. Lower limit More than c.f. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 3 8 17 29 15 6 2 10 20 30 40 50 60 70 3 11 28 57 72 78 80 0 10 20 30 40 50 60 80 77 69 52 23 8 2

Less than ogive and more than ogive combined graph,

From the graph, the perpendicular drawn from the point of intersection of two ogives meets x-axis at 34.14 (approx.)units from the origin. So, the required median of the given distribution is 34.14.

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