The branch of mathematics that deals with large amount of numerical data, is known as statistics. It is a science of gathering, tabulating, organizing, calculating and interpreting statistical data. Statistical methods are commonly used in researches and surveys where vast data are used. Generally, statistics do require different types of graphs in order to understand data more clearly. The most commonly used graphs are bar chart, histogram, frequency polygon, line graph, cumulative frequency curve etc.

Let us go ahead and understand about the **cumulative frequency curve **in this page. The cumulative frequency curve is also known as **Ogive**. Just to recall that the cumulative frequency of a variable is the summation of all the frequencies of variables previous to it. The frequency distribution of a given data set could be converted into a
cumulative frequency distribution by adding each frequency to the total
of the predecessors. An ogive is a curve that represents cumulative frequencies of the given variables. The graph of the cumulative frequency distribution is better known as the cumulative frequency curve or Ogive.

The word Ogive is basically a term used in the architecture to describe curves or curved shapes. Ogives are graphs that are used
to estimate how many numbers lie below or above a particular variable or value in a data.
In order to construct ogive, first the cumulative frequency of the variables is calculated using a frequency table. It is done by adding
the frequencies of all the previous variables in the given data set.
The result or the last number in the cumulative frequency table is always equal to the
total frequencies of the variables.

Ogive is a graph of a cumulative distribution, which shows data values on the horizontal axis and either the cumulative frequencies, the cumulative relative frequencies or cumulative percent frequencies on the vertical axis. The Ogive is constructed by plotting a point corresponding to the cumulative frequency of each class.

The most commonly used graphs of frequency distribution are as follows:

### Frequency Ogive

There are two ways of constructing an ogive or cumulative frequency curve. The steps for constructing less than Ogive chart and more than Ogive chart are given below:

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

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

**Steps for constructing a greater than or more than Ogive chart (more than Cumulative frequency curve):**

When 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

** Solution: **

First lets find the more than cumulative frequency corresponding to each class. 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.

Now we draw the horizontal and vertical axes and label them. Plot the cumulative frequencies corresponding to the lower limit of each class and join the points using a smooth curve.

The more than cumulative frequency curve is shown below.

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:

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.

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

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

### 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:**

(ii) Join the points by a smooth curve.

To reconstruct frequency distribution from cumulative frequency distribution.

**Question 2: **

** Solution: **

The most commonly used graphs of frequency distribution are as follows:

- Histogram
- Frequency polygon
- Frequency Curve
- Ogives (Cumulative frequency curves)

There are two ways of constructing an ogive or cumulative frequency curve. The steps for constructing less than Ogive chart and more than Ogive chart are given below:

- Draw and label the horizontal and vertical axes.
- Take the cumulative frequencies along the y axis (vertical axis) and the upper class limits on the x axis (horizontal axis)
- Plot the cumulative frequencies against each upper class limit.
- Join the points with a smooth curve.

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

When we write, 'less than 10 - less than 0', the difference gives the frequency 4 for the class interval (0 - 10) and so on.

- Draw and label the horizontal and vertical axes.
- Take the cumulative frequencies along the y axis (vertical axis) and the lower class limits on the x axis (horizontal axis)
- Plot the cumulative frequencies against each lower class limit.
- Join the points with a smooth curve.

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

When 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.

Class |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 | 60-70 |
70-80 | 80-90 |

F |
3 |
15 |
8 |
20 |
7 |
4 |
6 |
2 |

First lets find the more than cumulative frequency corresponding to each class. 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 class and join the points using a smooth curve.

The more than cumulative frequency curve is shown below.

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 curves are characterized by observations like equidistant from the central maximum and have the same frequency.

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

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

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

- In a J shaped or reverse J shaped frequency curve, the maximum occurs at one end or the other.
- U shaped frequency distribution curve has maxima at both the ends and a minimum in between.
- A multi-modal frequency curve has more than two maxima.
- A bi-modal frequency curve has two maxima.

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

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.

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

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

- The less than cumulative 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.
- 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.
- Less than cumulative frequency curve slope upwards from left to right whereas more than cumulative curve slope downwards from left to right.
- Ogives are the graphical representations used to find the Median of a frequency distribution.

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

**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.

(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.fTo reconstruct frequency distribution from cumulative frequency distribution.

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

Frequency distribution of the data:

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.

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

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

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