An Ogive (pronounced O-Jive) is a graph that represents the cumulative frequencies for the classes in a frequency distribution and it is a continuous frequency curve. Ogive has the shape of an elongated 'S' and is sometimes called a double curve with one portion being concave and the other being convex. While constructing, it is necessary to first have the frequency table. To plot an ogive, we need class boundaries and the cumulative frequencies. For grouped data, ogive is formed by plotting the cumulative frequency against the upper boundary of the class. For ungrouped data, cumulative frequency is plotted on the y-axis against the data which is on the x-axis.

Ogive is used to study the growth rate of data as it shows the accumulation of frequency.

## Ogive Graph

Representing data in the form of ogive curve makes it more effective to be understood as compared to arrangement that is represented using table.

Cumulative frequency table is also known as ogive.

Two different types of ogive can be drawn. They are less than type ogive and more then type ogive.

Following steps are necessary to plot a less than type ogive curve:
1. Start from the upper limit of class intervals and then, add class frequencies to get cumulative frequency distribution.
2. Take upper limit in the x-axis direction and cumulative frequencies along the y-axis direction.
3. Plot the points (x, y), where 'x' is the upper limit of a class and 'y' is the corresponding cumulative frequency.
4. Join the points by a smooth curve.

The less than type ogive looks like an elongated 'S' and starts at the lowest boundary on the axis and ends at the highest class boundary which corresponds to the total frequency of the distribution.

Following steps are necessary to plot a more than type ogive curve:

1. Start from the lower limits of class intervals and the total frequency is subtracted from the frequency to get the cumulative frequency distribution.
2. In the graph, consider the lower limit in the x-axis direction and cumulative frequencies along y-axis direction.
3. Plot the points (x, y), where 'x' is the lower limit of a class and 'y' is the corresponding cumulative frequency.
4. Join the points by a smooth curve.
More than type ogive looks like an elongated 'S' turned upside down.

### Solved Example

Question: For the data given below, construct a less than cumulative frequency table and plot its ogive.

 Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 -100 Frequency 3 5 6 7 8 9 10 12 6 4

Solution:
 Marks Frequency Less than cumulative frequency 0 - 10 3 3 10 - 20 5 8 20 - 30 6 14 30 - 40 7 21 40 - 50 8 29 50 - 60 9 38 60 - 70 10 48 70 - 80 12 60 80 - 90 6 66 90 - 100 4 70

Plot the points having abscissa as upper limits and ordinates as the cumulative frequencies (10, 3), (20, 8), (30, 14), (40, 21), (50, 29), (60,38), (70, 48), (80, 60), (90, 66), (100, 70) and join the points by a smooth curve.

## Ogive Example

Given below are some of the examples on ogive.

### Solved Examples

Question 1: For the data given below, construct a more than cumulative frequency table and plot its ogive.

 Marks 0 - 5 5 - 10 10 -15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50 Frequency 3 5 7 8 10 11 14 19 15 13

Solution:
 Marks Frequency More than cumulative frequency 0 - 5 3 95 5 - 10 5 95 - 3 = 92 10 -15 7 92 - 5 = 87 15 - 20 8 87 - 7 = 80 20 - 25 10 80 - 8 = 72 25 - 30 11 72 - 10 = 62 30 - 35 14 62 - 11 = 51 35 - 40 19 51 - 14 = 37 40 - 45 15 37 - 19 = 18 45 - 50 13 18 - 15 = 3

On the graph, plot the points (0, 95), (5, 92), (10, 87), (15, 80), (20, 72), (25, 62), (30, 51), (35, 37), (40, 18), (45, 3) and join the points by a smooth curve.

Question 2: Draw 'more than' and 'less than' ogive curves for the following data:

 Class Interval 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50 50 -55 55 - 60 60 - 65 65 -70 70 - 75 Frequency 2 5 8 10 13 17 20 16 12 18 19 20

Solution:
Let us calculate cumulative frequencies as follows:

 Class Interval Frequency Less than cumulative frequency More than cumulative frequency 15 - 20 2 2 155 20 - 25 5 2 + 5 = 7 155 - 2 = 153 25 - 30 8 7 + 8 = 15 153 - 5 =  148 30 - 35 10 15 + 10 = 25 148 - 8 = 140 35 - 40 13 25 + 13 = 38 140 - 10 = 130 40 - 45 17 38 + 17 =  55 130 - 13 = 117 45 - 50 20 55 + 20 =  75 117 - 17 = 100 50 -55 16 75 + 16 =  91 100 - 20 = 80 55 - 60 12 91 + 12 = 103 80 -16 = 64 60 - 65 15 103 + 15 =  118 64 - 12 =  52 65 - 70 17 118 + 17 =  135 49 - 15 = 37 70  - 75 20 135 + 20 = 155 32 - 17 = 20

Less than ogive:
For less than ogive, plot the points, (20, 2), (25, 7), (30, 15), (35, 25), (40, 38), (45, 55), (50, 75), (55, 91), (60, 103), (65, 118), (70, 135), (75, 155) and join the points by a smooth curve.

Less than ogive plot for the given data is as follows:

More than ogive:
For more than ogive, plot the points, (15, 155), (20, 153), (25, 148), (30, 140), (35, 130), (40, 117), (45, 100), (50, 80), (55, 64), (60, 52), (65, 37), (70, 20) and join the points by a smooth curve.

More than ogive plot for the given data is as follows: