Graphical Representation of Cumulative Frequency Distribution: A frequency distribution is a graphical or tabular representation of the frequency of repeated items. It displays the frequency of items or the number of times they occur in a graphical format. It is used to arrange the data in a tabular format. A cumulative frequency distribution is the sum of the initial frequency and all frequencies below it in a frequency distribution.

An ogive is ahand graph showing the curve of a cumulative frequency distribution. This article will study the graphical representation of cumulative frequency distribution called a cumulative frequency curve or an ogive.

What is Frequency Distribution?

The collected data is organised in table form using frequency distribution. Students’ marks, temperatures in some cities, points earned in a match, and so on could be included in the data. After gathering data, we must present it in a meaningful way for better comprehension. Organise the data such that all of its characteristics are summarised in a table.

Let us consider an example. The following are the scores of \(20\) students in an exam.

\(10,15,17,20,15,10,20,17,14,14,15,17,20,10,13,15,13,13,17,10.\) Let us present this data in a table and determine the frequency of the students who got the same marks.

Marks Obtained	Number of students
\(10\)	\(4\)
\(13\)	\(3\)
\(14\)	\(2\)
\(15\)	\(4\)
\(17\)	\(4\)
\(20\)	\(3\)

All of the collected data is arranged under the marks obtained column and the number of students column. This makes it simple to understand the information provided, and we can see the number of students that received the same grade.

As a result, frequency distribution in statistics assists us in organising data in a way that makes it simple to understand its characteristics at a glimpse.

Learn About Cumulative Frequency Distribution

Frequency Distribution Graph

Another technique to present data in the form of graphs is to use a frequency distribution graph. The graphs make it simple to understand the collected data.

Bar Graphs: Rectangular bars with consistent width and identical spacing between the rectangular bars display data in bar graphs.

Histogram: A histogram is a graphical representation of data that uses rectangular bars of varying heights to represent the data. There is no gap between the rectangular bars in a histogram.

Pie Chart: A pie chart is a graph that uses a circular chart to display data visually. It records data in a circular pattern and then divides it into sectors, displaying a subset of the total data.

Frequency Polygons: The mid-points of the bars in a histogram are joined to form a frequency polygon.

Types of Frequency Distribution

The four different types of frequency distribution are:

1. Ungrouped Frequency Distribution
2. Grouped Frequency Distribution
3. Relative Frequency Distribution
4. Cumulative Frequency Distribution

In this article, let us study the concept of graphical representation of cumulative frequency distribution.

Cumulative Frequency Distribution

Cumulative Frequency: In a discrete frequency distribution, the cumulative frequency of a particular value of the variable is the total of all the frequencies of the values of the variable which are less than or equal to the particular value.

For example, the following table gives the frequency distribution of the number of children in \(10\) families. If \(X\) denotes the number of children in a family, then the cumulative frequency of \(X = 2\) is \(11.\) Because there are \(11\) families in which the number of children is either \(1\) or \(2.\) The cumulative frequency of \(X = 3\) is \(15,\) which means that there are \(15\) families in which the number of children is \(3\) or less. Similarly, cumulative frequencies of \(X = 4\) and \(X = 5\) are \(18\) and \(20,\) respectively.

Number of children	Number of families (Frequency)	Cumulative Frequency (C.F.)
\(1\)	\(5\)	\(5\)
\(2\)	\(6\)	\(11\)
\(3\)	\(4\)	\(15\)
\(4\)	\(3\)	\(18\)
\(5\)	\(2\)	\(20\)
Total	\(20\)

In a grouped frequency distribution, the cumulative frequency of a class is the total of all frequencies up to and including that particular class.

Cumulative Frequency Distribution: A table showing how cumulative frequencies are distributed over various classes is called a cumulative frequency distribution table.

Graphical Representation of Cumulative Frequency Distribution

Let us learn about the construction of cumulative frequency polygon and cumulative frequency curves or ogives.

Drawing the cumulative frequency polygons and cumulative frequency curves or ogives is more or less the same. The only difference is that in simple frequency curves and polygons, the frequencies are plotted against class marks of the class intervals. In contrast, in the case of a cumulative frequency polygon or curves, the cumulative frequencies are plotted against the lower or upper limits of the class intervals depending upon how the series has been cumulated. There are two methods of constructing a frequency polygon and an ogive. Let us now discuss the two methods.

1. Less than method
2. More than method

Less Than Method

To construct a cumulative frequency polygon and an ogive by less than method, we use the following steps:

• 1. Start with the upper limits of class intervals and add class frequencies to obtain the cumulative frequency distribution.
• 2. Mark upper-class limits along \(X\)-axis on a suitable scale.
• 3. Mark cumulative frequencies along \(Y\)-axis on a suitable scale.
• 4. Plot the points \(\left({{x_i},{f_i}} \right)\) where \({x_i}\) is the upper limit of a class and \({f_i}\) is the corresponding cumulative frequency.
• 5. Join the points obtained in the above step by a hand smooth curve to get the ogive, and to get the cumulative frequency polygon, join the points obtained in the above step by line segments.

More Than Method

To construct a cumulative frequency polygon and an ogive by more than method, we use the following steps:

• 1. Start with the lower limits of the class intervals, and from the total frequency, subtract the frequency of each class to obtain the cumulative frequency distribution.
• 2. Mark the lower class limits along \(X\)-axis on a suitable scale.
• 3. Mark the cumulative frequencies along \(Y\)-axis on a suitable scale.
• 4. Plot the points \(\left({{x_i},{f_i}} \right)\) where \({x_i}\) is the upper limit of a class and \({f_i}\) is the corresponding cumulative frequency.
• 5.Join the points obtained in the above step by a hand smooth curve to get the ogive and get the cumulative frequency polygon to join these points by line segments.

Learn About Frequency Distribution Table

Solved Examples: Graphical Representation of Cumulative Frequency Distribution

Q.1. Draw an ogive for the following frequency distribution by less than method.

Marks	Number of students
\(0 – 10\)	\(7\)
\(10 – 20\)	\(10\)
\(20 – 30\)	\(23\)
\(30 – 40\)	\(51\)
\(40 – 50\)	\(6\)
\(50 – 60\)	\(3\)

Ans: We first prepare the cumulative frequency distribution table by less than method as given below:

Marks	Number of students	Marks Less than	Cumulative Frequency
\(0 – 10\)	\(7\)	\(10\)	\(7\)
\(10 – 20\)	\(10\)	\(20\)	\(17\)
\(20 – 30\)	\(23\)	\(30\)	\(40\)
\(30 – 40\)	\(51\)	\(40\)	\(91\)
\(40 – 50\)	\(6\)	\(50\)	\(97\)
\(50 – 60\)	\(3\)	\(60\)	\(100\)

Now, we mark the upper-class limits along \(x\)-axis on a suitable scale and the cumulative frequencies along \(y\)-axis on a suitable scale.

Thus, we plot points \(\left({0,0} \right),\left({10,7} \right),\left({20,17} \right),\left({30,40} \right),\left({40,91} \right),\left({50,97} \right)\)) and \(\left({60,100} \right).\)

Now, we join the plotted points by ahand curve to obtain the required ogive.

Q.2. Following is the age distribution of a group of students. Draw the cumulative frequency curve of less than type.

Age (in years)	Frequency
\(4 – 5\)	\(36\)
\(5 – 6\)	\(42\)
\(6 – 7\)	\(52\)
\(7 – 8\)	\(60\)
\(8 – 9\)	\(68\)
\(9 – 10\)	\(84\)
\(10 – 11\)	\(96\)
\(11 – 12\)	\(82\)
\(12 – 13\)	\(66\)
\(13 – 14\)	\(48\)
\(14 – 15\)	\(50\)
\(15 – 16\)	\(16\)

Ans: We will prepare the less than type cumulative frequency table as shown below.

Age (in years)	Cumulative Frequency
Less than \(5\)	\(36\)
Less than \(6\)	\(78\)
Less than \(7\)	\(130\)
Less than \(8\)	\(190\)
Less than \(9\)	\(258\)
Less than \(10\)	\(342\)
Less than \(11\)	\(438\)
Less than \(12\)	\(520\)
Less than \(13\)	\(586\)
Less than \(14\)	\(634\)
Less than \(15\)	\(684\)
Less than \(16\)	\(700\)

We plot the points \(\left({5,36} \right),\left({6,78} \right),\left({7,130} \right),\left({8,190} \right),\left({9,258} \right),\left({10,342} \right),\left({11,438} \right),\left({12,520} \right),\left({13,586} \right),\)

\(\left({14,634} \right),\left({15,684} \right)\) and \(\left({16,700} \right).\)

Now, we join the plotted points by ahand curve to obtain the required ogive.

Q.3. For the following frequency distribution, draw a cumulative frequency curve of the more than type and obtain the median value.

Class interval	Frequency
\(0 – 10\)	\(5\)
\(10 – 20\)	\(15\)
\(20 – 30\)	\(20\)
\(30 – 40\)	\(23\)
\(40 – 50\)	\(17\)
\(50 – 60\)	\(11\)
\(60 – 70\)	\(9\)

Ans: We will prepare the more than type cumulative frequency table as shown below.

More than \(60\)	\(9\)
More than \(50\)	\(20\)
More than \(40\)	\(37\)
More than \(30\)	\(60\)
More than \(20\)	\(80\)
More than \(10\)	\(95\)
More than \(5\)	\(100\)

We plot the points \(\left({5,100} \right),\left({10,95} \right),\left({20,80} \right),\left({30,60} \right),\left({40,37} \right),\left({50,20} \right),\left({60,9} \right)\)

Now, we join the plotted points by ahand curve to obtain the required ogive.

Here, \(N = 100 \Rightarrow \frac{N}{2} = 50\)

So, from \(P\left({0,50} \right),\) draw \(PQ||x\)-axis, meeting the curve at \(Q.\) Draw \(QM \bot OX,\) meeting \(x\)-axis at \(M\) whose coordinates are \(\left({35,0} \right).\)

Hence median \( = 35\)

Q.4. The annual profits of a shopping complex shops in a locality give rise to the following distribution. Draw both ogives for the data above. Hence obtain the median profit.

Profit (\(₹\) in lakhs)	Number of shops
More than or equal to \(5\)	\(30\)
More than or equal to \(10\)	\(28\)
More than or equal to \(15\)	\(16\)
More than or equal to \(20\)	\(14\)
More than or equal to \(25\)	\(10\)
More than or equal to \(30\)	\(7\)
More than or equal to \(35\)	\(3\)

Ans: We first draw the coordinate axes, with lower limits of the profit along the horizontal axis and the cumulative frequency along the vertical axes. Then, we plot the points \(\left({5,30} \right),\left({10,28} \right),\left({15,16}\right),\left({20,14}\right),\left({25,10} \right),\left({30,7} \right)\) and \(\left({35,3} \right).\) We join these points with a smooth curve to get the ‘more than’ ogive, as shown below.

Classes	Number of shops	Cumulative Frequency
\(5 – 10\)	\(2\)	\(2\)
\(10 – 15\)	\(12\)	\(14\)
\(15 – 20\)	\(2\)	\(16\)
\(20 – 25\)	\(4\)	\(20\)
\(25 – 30\)	\(3\)	\(23\)
\(30 – 35\)	\(4\)	\(27\)
\(35 – 40\)	\(3\)	\(30\)

Using these values, we plot the points \(\left({10,2} \right),\left({15,14} \right),\left({20,16} \right),\left({25,20} \right),\left({30,23} \right),\left({35,27} \right),\left({40,30} \right)\) on the same axes as in above graph to get the ‘less than’ ogive, as shown in below figure.

The abscissa of their point of intersection is nearly \(17.5,\) which is the median, which can also be verified by using the formula.

Hence, the median profit (in lakhs) is \(₹17.5.\)

Q.5. The following table gives production yield per hectare of wheat of 100 farms of a village:

Production yield (kg/ha)	Number of farms
\(40 – 45\)	\(4\)
\(45 – 50\)	\(6\)
\(50 – 55\)	\(16\)
\(55 – 60\)	\(20\)
\(60 – 65\)	\(30\)
\(65 – 70\)	\(24\)

Change the distribution to a ‘more than’ type distribution and draw its ogive.

Ans: We will prepare the more than type cumulative frequency table as shown below.

More than \(65\)	\(24\)
More than \(60\)	\(54\)
More than \(55\)	\(74\)
More than \(50\)	\(90\)
More than \(45\)	\(96\)
More than \(40\)	\(100\)

We plot the points \(\left({40,100} \right),\left({45,96} \right),\left({50,90} \right),\left({55,74} \right),\left({60,54}\right),\left({65,24} \right)\)

Now, we should join the plotted points by ahand curve to obtain the required give.

Summary

In this article, we have learnt the definition of frequency distribution and cumulative frequency. Also, we have learnt the meaning of the frequency distribution table and the graphical representation of cumulative frequency and solved some example problems.

FAQs on Graphical Representation of Cumulative Frequency Distribution

Q.1. What is a frequency distribution?
Ans: A frequency distribution is a graphical or tabular representation of the frequency of repeated items. It displays the frequency of items or the number of times they occurred in a graphical format. It is used to arrange the data in a tabular format.

Q.2. Define cumulative frequency distribution.
Ans: A cumulative frequency distribution is the sum of the initial frequency and all frequencies below it in a frequency distribution.

Q.3. What is more than type ogive?
Ans: The frequency of a class is increased by adding the frequencies of the subsequent classes. The cumulative series is sometimes known as the more than or greater than series. It’s made by deducting the first-class frequency from the total, then subtracting the second class frequency from that, and so on. The cumulative series is bigger than or equal to the upward cumulation result.

Q.4. What is less than type ogive?
Ans: The frequency of a class is increased by adding the frequencies of all preceding classes. The less than cumulative series is the name given to this set of numbers. It’s made by adding the first-class frequency by the second-class frequency, then by the third-class frequency, and so forth. The less-than-cumulative series is the consequence of the downward cumulation.

Q.5. What is an ogive in statistics?
Ans: A cumulative frequency polygon, often known as an ogive, is a frequency polygon that displays cumulative frequencies. In other words, the cumulative percents are added from left to right on the graph. Cumulative frequency is plotted on the \(y\)-axis, while class boundaries are plotted on the \(x\)-axis in an ogive graph.

We hope this detailed article on the graphical representation of cumulative frequency distribution helped you in your studies. If you have any doubts, queries, or suggestions regarding this article, feel to ask us in the comment section, and we will be more than happy to assist you. Happy learning!