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Ogive Definition

Ogive Chart

Types of FrequencyCurves

Ogive Graph

Ogive Example

elated Concepts

lculate Cumulative Frequency

equency

nomial Cum ulative Dis tribution

mulative Normal Dis tribution

mulative Poisson Distribution

mulative Probability Distribution

Ogive
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Ogive Chart Back to Top

4. Ogives (Cumulative frequency curves)

Ogive is defined as the frequency distribution graph of a series. Such graphs are more appealing to eye

than the tabulated data and help us to facilitate comparative study.

An Ogive Chart is a curve of the cumulative frequency distribution or cumulative relative frequency

distribution.To draw such a curve, first of all the s imple frequency must be expressed as percentage of the total

frequency. Then, such percentages are cumulated and plotted as in the case of an ogive.

Frequency Ogive

There are two ways of constructing an ogive or cumulative frequency curve. The steps for constructing less

han Ogive chart and more than Ogive chart are given below:

Steps for constructing a less than Ogive chart (less than Cumulative frequency curve):

1. Draw and label the horizontal and vertical axes.

2. Take the cumulative frequencies along the y axis (vertical axis) and the upper class limits on the x

axis (horizontal axis)

3. Plot the cumulative frequencies against each upper class limit.4. Join the points with a smooth curve.

Let us see with the help of a table how to construct a 'less than' Ogive chart:

hen we write, 'less than 10 - less than 0', the difference gives the frequency 4 for the class interval (0 - 10)

and so on.

Steps for constructing a greater than or more than Ogive chart (more than Cumulative frequency

curve):

1. Draw and label the horizontal and vertical axes.

2. Take the cumulative frequencies along the y axis (vertical axis) and the lower class limits on the x

axis (horizontal axis)

3. Plot the cumulative frequencies against each lower class limit.

4. Join the points with a smooth curve.

Let us see with the help of a table how to construct a 'more than' Ogive chart:
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hen we write 'more than 0 - more than 10', the difference gives the frequency 4 for the class interval (0 - 10)

and so on.

Corresponding to the point of intersection of less than cumulative frequency curve, greater

than or more than cumulative frequency curve is the Median of the distribution. So, we can find

the middlemost value of the series if we draw the less than and greater than Ogives.

Solved Example

Question: Draw the more than cumulative frequency curve for the following data

Class 10-20 20-30 30-40 40-50 50-6060-70 70-80 80-90

F 3 15 8 20 7 4 6 2

Solution:

First lets find the more than cumulative frequency corresponding to each c lass. For this the

frequencies of the succeeding classes are added to the frequency of a class . The greater

than cumulative frequency table is given below.

Lower limit Frequency More than Cumulative Frequency

10 3 65

20 15 65 - 3 = 62

30 8 62 - 15 = 47

40 20 47 - 8 = 41

50 7 41 - 20 = 19

60 4 19 - 7 = 12

70 6 12 - 4 = 8

80 2 8 - 6 = 2

Now we draw the horizontal and vertical axes and label them. Plot the cumulative frequencies

corresponding to the lower limit of each c lass and join the points using a smooth curve.

The more than cumulative frequency curve is shown below.
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Types of Frequency Curves Back to Top

Ogive Graph Back to Top

Frequency curves made into practice take on certain characteristic shapes and they are (a) Symmetrical or

bell shaped (b) Skewed to right (c) Skewed to left (d) Uniform.

Symmetrical or bell shaped:

Symmetrical or bell shaped curves are characterized by observations like equidistant from the central

maximum and have the same frequency.

Skewed to Right:

Curves that have a tail to the left are said to be skewed to the left.

Skewed to Left:

Curves that have tails to the right are said to be skewed to the right.

Uniform:

Curves that have approximately equal frequencies all across are said to be uniformly distributed.

Other curves:

1. In a J shaped or reverse J shaped frequency curve, the maximum occurs at one end or the other.

2. U shaped frequency distribution curve has maxima at both the ends and a minimum in between.

3. A multi-modal frequency curve has more than two maxima.

4. A bi-modal frequency curve has two maxima.

A frequency dist ribution when cumulated, we get a cumulative frequency distribution. There are two methods of

cumulating a series. Graphs are used to represent the frequency distributions or to represent the relationship

between two variables. The most commonly used type of graph is a graph of frequency distribution.

Graphs of Frequency Distribution

The graphs of frequency distribution are frequency graphs that are used to reveal the characteristics of discrete

and continuous data. Such graphs are more appealing to the eye than the tabulated data. This helps us to

facilitate the comparative study of two or more frequency distributions. We can compare the shape and pattern

of the two frequency distributions.

The two methods of Ogives are less than Ogive and greater than or more than Ogive.

First Method:

The frequencies of all preceding classes are added to the frequency of a class. This series is called the

less than cumulative series.
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Ogive Example Back to Top

It is constructed by adding the first class frequency to the second class frequency and then to the third class

frequency and so on. The downward cumulation result in the less than cumulative series.

Second Method :

The frequencies of the succeeding classes are added to the frequency of a class. This series is called as

the more than or greater than cumulative series.

It is constructed by subtracting the first class second class frequency from the total, third class frequency from

hat and so on. The upward cumulation result is greater than or more than the cumulative series.

hen the graphs of these series are drawn, we get a cumulative frequency curve or Ogives.

1. The less than cumulat ive frequency curve is known as Less than Ogive and the greater

than cumulative frequency curve is known as the Greater than or More than Ogive.

2. Less than Ogive curves are obtained by plotting less than cumulative frequencies

against the upper limits of each class interval whereas more than Ogive curves are

obtained by plotting more than cumulative frequencies against the lower limits of each

class interval.

3. Less than cumulative frequency curve slope upwards from left to right whereas more

than cumulative curve slope downwards from left to right.

4. Ogives are the graphical representat ions used to find the Median of a frequency

distribution.

Given below are some of the examples on Ogive.

Solved Examples

Question 1: Using the data given below, construct a 'more than' cumulative frequency table and

draw the Ogive.

Solution:

Step 1:

'more than' cumulative frequency table for the given data:

Step 2:

To plot an Ogive:

(i) We plot the points with coordinates having abscissae as actual lower limits and ordinates

as the cumulative frequencies,

(70.5, 2), (60.5, 7), (50.5, 13), (40.5, 23), (30.5, 37), (20.5, 49),(10.5, 57), (0.5, 60) are the

coordinates of the points.

(ii) Join the points by a smooth curve.
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(iii) An Ogive is connected to a point on the X-axis representing the actual upper limit of the

last class [in this case) i.e., point (80.5, 0)].

Step 3:

Scale:

X-axis 1 cm = 10 marks

Y-axis 2 cm = 10 c.f

To reconstruct frequency distribution from cumulative frequency distribution.

Question 2:

Draw a 'less than' ogive curve for the following data:

Solution:

Frequency distribution of the data:


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To plot an Ogive:

(i) We plot the points with coordinates having abscissa as actual limits and ordinates as the

cumulative frequencies, (10, 2), (20, 10), (30, 22), (40, 40), (50, 68), (60, 90), (70, 96) and (80,

100) are the coordinates of the points.

(ii) Join the points plotted by a smooth curve.

(iii) An Ogive is connected to a point on the X-axis representing the actual lower limit of the

first class.

Scale:

X -axis 1 cm = 10 marks, Y -axis 1cm = 10 c.f.

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