Ogive (Cumulative Frequency Curve) and its Types
A method of presenting data in the form of graphs that provides a quick and easier way to understand the trends of the given set of data is known as Graphic Presentation. The two types of graphs through which a given set of data can be presented are Frequency Distribution Graphs and Time Series Graphs. The four most common graphs under Frequency Distribution Graphs are Line Frequency Graph, Histogram, Frequency Polygon, Frequency Curve, and Ogive.
What is Ogive?
An Ogive or Cumulative Frequency Curve is a curve of a data set obtained by an individual through the representation of cumulative frequency distribution on a graph. As there are two types of cumulative frequency distribution; i.e., Less than cumulative frequencies and More than cumulative frequencies, the ogives are also of two types:
1. Less than Ogive
2. More than Ogive
Less than Ogive
The steps required to present a less than ogive graph are as follows:
Step 1: To present a less than ogive graph, add the frequencies of all the preceding class intervals to the frequency of a class.
Step 2: After that, plot the less than cumulative frequencies on the Y-axis against the upper limit of the corresponding class interval on the X-axis.
Step 3: In the last step, join these points by a smooth freehand curve, which is the resulting less than ogive.
A less than ogive curve is an increasing curve that slopes upwards from left to right.
Example:
Draw a ‘less than’ ogive curve from the following distribution of the marks of 50 students in a class.
Marks 10-20
20-30
30-40
40-50
50-60
60-70
70-80
No. of Students 6
4
15
5
8
7
5
Solution:
First of all, we have to convert the frequency distribution into a less than cumulative frequency distribution.
Marks
No. of Students (f)
No. of Students (cf)
Less than 20
6
6
Less than 30
4
6 + 4 = 10
Less than 40
15
6 + 4 + 15 = 25
Less than 50
5
6 + 4+ 15 + 5 = 30
Less than 60
8
6 + 4 + 15 + 5 + 8 = 38
Less than 70
7
6 + 4 + 15 + 5 + 8 + 7 = 45
Less than 80
5
6 + 4 + 15 + 5 + 8 + 7 + 5 = 50
Now, plot these values of cumulative frequency on a graph.
More than Ogive
The steps required to present a more than ogive graph are as follows:
Step 1: To present a more than ogive graph, add the frequencies of all the succeeding class intervals to the frequency of a class.
Step 2: After that, plot the more than cumulative frequencies on the Y-axis against the upper limit of the corresponding class interval on the X-axis.
Step 3: In the last step, join these points by a smooth freehand curve, which is the resulting more than ogive.
A more than ogive curve is a decreasing curve that slopes downwards from left to right.
Example:
Draw a ‘more than’ ogive curve from the following distribution of the marks of 50 students in a class.
Marks 10-20
20-30
30-40
40-50
50-60
60-70
70-80
No. of Students 6
4
15
5
8
7
5
Solution:
First of all, we have to convert the frequency distribution into a more than cumulative frequency distribution.
Marks
No. of Students (f)
No. of Students (cf)
More than 10
6
5 + 7 + 8 + 5 + 15 + 4 + 6 = 50
More than 20
4
5 + 7 + 8 + 5 + 15 + 4 = 45
More than 30
15
5 + 7 + 8 + 5 + 15 = 40
More than 40
5
5 + 7 + 8 + 5 = 25
More than 50
8
5 + 7 + 8 = 20
More than 60
7
5 + 7 = 12
More than 70
5
5
Now, plot these values of cumulative frequency on a graph.
Both ‘Less Than’ and ‘More Than’ Ogives
Both the ‘less than’ and the ‘more than’ ogives can be plotted on the same graph, and the point at which these two curves intersect is the median of the given data set.
Example:
Draw both ‘less than’ and ‘more than’ ogive curve from the following distribution of the marks of 50 students in a class.
Marks 10-20
20-30
30-40
40-50
50-60
60-70
70-80
No. of Students 6
4
15
5
8
7
5
Solution:
First of all, we have to convert the frequency distribution into a less than and more than cumulative frequency distribution.
Marks
No. of Students (cf)
Marks
No. of Students (cf)
Less than 20
6
More than 10
50
Less than 30
10
More than 20
45
Less than 40
25
More than 30
40
Less than 50
30
More than 40
25
Less than 60
38
More than 50
20
Less than 70
45
More than 60
12
Less than 80
50
More than 70
5
Now, plot these values of less than and more than cumulative frequency on a graph.
Ogive (Cumulative Frequency Curve) – FAQs
What is an ogive?
An ogive, also known as a cumulative frequency curve, is a graph that represents the cumulative frequencies for a dataset. It shows the running total of frequencies, helping to understand the distribution of data.
What is the purpose of an ogive?
Ogives are used to analyze cumulative distributions, such as cumulative income, cumulative expenditure, and cumulative production. They help in understanding how total values accumulate over the range of data.
How can an ogive be used to find the median?
The median can be found using an ogive by locating the point where 50% of the data is accumulated. For a “less than” ogive, find the value on the x-axis corresponding to the 50% cumulative frequency mark.
How do you interpret an ogive?
An ogive shows how cumulative frequencies build up across classes or intervals. A steeper slope indicates a higher frequency in that interval, while a flatter slope indicates a lower frequency.
Can ogives be used for both discrete and continuous data?
Yes, ogives can be used for both discrete and continuous data. For discrete data, plot cumulative frequencies against each data point. For continuous data, plot cumulative frequencies against class boundaries.
How can ogives be used to compare two datasets?
To compare two datasets using ogives, plot both cumulative frequency curves on the same graph. Differences in the curves will highlight differences in the distributions, such as variations in median, spread, and skewness.
What is the relationship between a histogram and an ogive?
A histogram shows the frequency distribution of a dataset, while an ogive shows the cumulative frequency distribution. The ogive can be derived from the histogram by summing frequencies cumulatively.