Cumulative Frequency

Cumulative Frequency: In statistics, cumulative frequency is defined as the sum of frequencies distributed across various class intervals. This involves organizing the data and their totals into a table where the frequencies are allocated according to each class interval.

In this article, we will cover a thorough explanation of cumulative frequency, cumulative frequency curve, formula, and a few examples based on it for better understanding.

What is Cumulative Frequency?

Cumulative Frequency is the sum of all previous frequencies of the given data. Consider the frequency of the first class interval plus the frequency of the second class interval, the sum is then added to the frequency of the third class interval, and so on, to get the cumulative frequency. Therefore, the cumulative frequency table, also known as the cumulative frequency distribution, is the table that shows the cumulative frequencies that are split over various classes.

In most cases, the number of observations in the given data set that fall above or below a specific frequency is determined using the cumulative frequency distribution.

Cumulative Frequency Meaning

Cumulative frequency (c.f.) is the term used to determine the total number of observation above the current point, irrespective of the class of the given dataset. In simple terms, it is the sum of all previous observation of the given data.

Cumulative frequency is classified into two types: lesser than type and greater than type. The number of observations in a given data set that fall above (or below) a specific frequency is determined using cumulative frequency.

Cumulative Frequency Formula

The frequencies of all the data points that are equal to or less than a given value are added up to determine the cumulative frequency (CF) at that particular data point. In terms of math, it is expressed as:

[Tex]CF_i = \sum_{j=1}^{i} f_j [/Tex]

​where,

• CF_i is Cumulative Frequency at the i^th Data Point
• F_j is the j^th Data Point’s Frequency

Cumulative Frequency Distribution

Cumulative Frequency Distribution is a technique used in statistics for data organization and analysis. To create a cumulative frequency table, the frequencies of each data point are added up to a specific value. This table makes it easy to understand how many observations in the dataset fall below or equal to a given point.

When comparing various data sets, determining central tendencies, and visualizing the overall pattern of the data, the cumulative frequency distribution is especially helpful. It is frequently used in statistical analysis to learn more about the properties and distribution of a dataset.

Types of Cumulative Frequency

Cumulative Frequency are classified into two types namely:

• Less than Cumulative Frequency
• More than Cumulative Frequency

Less than Cumulative Frequency

Less than cumulative frequency, also known as a less than ogive, is a rising curve. It is obtained by adding the first-class frequency to the second-class frequency, and so on. Here, the cumulate begins from the lowest to the highest class.

In this frequency curve, the points are plotted using upper limits (x-axis) and their corresponding cumulative frequency (y-axis).

More than Cumulative Frequency

More than cumulative frequency, also known as a greater than cumulative frequency, is a downward curve. It is obtained by determining the cumulative total frequencies starting from the last class to the first class. Here, the cumulate begins from the highest to the lowest class.

In this frequency curve, the points are plotted using lower limits (x-axis) and their corresponding cumulative frequency (y-axis).

How to Calculate Cumulative Frequency?

Cumulative frequency is a statistical method that can be used to organize and examine data. It involves adding up all of the values of frequencies cumulatively within a dataset. Here’s a step-by-step explanation to find the cumulative frequency of the given dataset:

Step 1: Organize Data

To get started, form the dataset into ascending or descending order.

Step 2: Create a Frequency Table

Make a frequency table with the Value column and Frequency column then List all unique values in the dataset and their corresponding frequency.

Step 3: Add Cumulative Frequencies

Create a new column labeled “Cumulative Frequencies” in the frequency table. The cumulative frequency for the first row is the same as the frequency. Add the current row’s frequency to the previous row’s cumulative frequency for each subsequent row.

Example: Creating a Cumulative Frequency table with the below values and frequencies

Values	10	15	20	25	30
Frequency	3	2	4	1	5

Solution:

Start by arranging data in ascending order and write the corresponding frequency of each in a table. Then find the cumulative frequency of each by adding the frequency of the previous observation/value.

Value	Frequency	Cumulative Frequency
10	3	3
15	2	3+2=5
20	4	5+4=9
25	1	9+1=10
30	5	10+5=15

In this example, the cumulative frequency represents the sum of the frequencies up to a specific value.

Cumulative Frequency Table

A tabular representation of data that shows the cumulative frequencies for every value in a dataset is called a cumulative frequency table. The following steps will help you make a cumulative frequency table:

Example: Creating a Cumulative Frequency table with the below values and frequencies

Score: 60, 70, 80, 90, 100

Frequency: 5, 8, 12, 7, 3

Solution:

Below is the Cumulative frequency table of the above-given data:

Score	Frequency	Cumulative Frequency
60	5	5
70	8	5+8=13
80	12	13+12=25
90	7	25+7=32
100	3	32+3=35

Read More: Frequency Distribution Table

Cumulative Frequency Curve

A cumulative frequency curve, also known as an ogive, graphically depicts the cumulative frequency distribution of a dataset. Plotting the cumulative frequencies against the upper-class boundaries or the midpoints of the class interval yields a smooth curve.

To generate a cumulative frequency curve, perform the subsequent steps:

• Step 1: Put the data in ascending order in your dataset.
• Step 2: To acquire cumulative frequency, gradually add up the frequencies.
• Step 3: Determine the axes’ scale by consulting your data.
• Step 4: For every data value and its cumulative frequency, plot points on the graph.
• Step 5: Connect the spots with a gentle freehand curve.
• Step 6: Put “Data Values” on the x-axis and “Cumulative Frequency” on the y-axis. Put a heading on the chart.

Cumulative frequency curve can be further plotted in two ways:

• Less than cumulative frequency Curve
• More than cumulative frequency Curve

Example: Draw a Less than and More than Cumulative Frequency Curve for the below given data:

Class intervals	0-10	10-20	20-30	30-40	40-50
Frequency (Students)	5	8	12	15	20

Solution:

Calculating the cumulative frequency for less than curve:

Class intervals	Frequency	Cumulative Frequency
less than 10	5	5
less than 20	8	13
less than 30	12	25
less than 40	15	40
less than 50	20	60

Now, we graph the cumulative frequencies about the class interval upper bound.

Now, calculate the cumulative frequency for more than one curve:

Class Intervals	Frequency	Cumulative Frequency
more than 0	–	60
more than 10	5	60-5=55
more than 20	8	55-8=47
more than 30	12	47-12=35
more than 40	15	35-15=20
more than 50	20	20-20=0

Now, we graph the cumulative frequencies about the class interval lower bound.

Cumulative Frequency Polygon

A graphical depiction of the cumulative frequencies in a dataset is called a cumulative frequency polygon. Plotting points that reflect the cumulative frequencies at the upper-class borders of the intervals in a grouped dataset and joining these points by line segment results in this polygon.

The cumulative frequency polygon can also be created in two ways

• Less than cumulative frequency polygon
• More than cumulative frequency polygon

Note: The only difference between a Cumulative Frequency Curve and a Cumulative Frequency Polygon is that the curve is drawn freehand and in a polygon line segments are used to connect the plotted coordinates.

Read: Frequency Polygon

Cumulative Frequency Graph

A cumulative frequency histogram is a bar-based graphical depiction of a cumulative frequency distribution. It combines the features of a histogram with a cumulative frequency polygon. The following procedures will help you create a cumulative frequency histogram:

• Arrange Data: To begin, take a set of data and divide it up into class intervals.
• Compute Cumulative Frequencies: The cumulative frequency for each interval is the total of all the frequencies up to and including that interval.
• Build the Histogram: To build the Histogram, drawbars for each interval, where the height of the bar corresponds to the interval’s frequency.
• Create Cumulative Frequency Curve: Plot points for each interval beside the histogram, with the x-coordinate representing the interval’s upper-class boundary and the y-coordinate representing the cumulative frequency. To create a cumulative frequency curve, join these points.

Relative Frequency

Relative frequency indicates how frequently a specific value or event occurs about the total number of observations. It is frequently stated as a percentage or fraction. The following formula can be used to get relative frequency:

Relative Frequency = Frequency of a Value or Event / Total Number of Observations ​

In mathematical terms, if f_i​ represents the frequency of a specific value or event, and N is the total number of observations, the relative frequency (RF_i​) is given by:

RF_i​ = f_i / N

Relative Cumulative Frequency

Relative cumulative frequency is an expansion of relative frequency. It shows the relative frequencies’ cumulative (running) total up to a given value. The following formula can be used to get relative cumulative frequency:

[Tex]\text{Relative Cumulative Frequency} = \sum_{i=1}^{n} \text{Relative Frequency}_i   [/Tex]​

To put it another way, you total up each of the distinct relative frequencies from the first to the n^th observation.

For example, Find the relative cumulative frequency for the table added below,

Marks	10	20	30	40	50
Number of students	6	9	4	3	1

The relative cumulative frequency table for the same is added below,

Score	Frequency	Relative Frequency	Relative Cumulative Frequency
10	6	6/24 = 0.25	0.25
20	9	9/24 = 0.375	0.25 + 0.375 = 0.625
30	4	4/24 = 0.1667	0.625 + 0.1667 = 0.7917
40	3	3/24 = 0.125	0.7917 + 0.125 = 0.9167
50	1	1/24 = 0.041667	0.9167 + 0.041667 = 1(approx)

People Also Read:

• Frequency and Related Frequency
• Relative Frequency
• Graphical Representation of Data

Cumulative Frequency Solved Examples

Various examples on Cumulative frequency are,

Example 1: Below is the data of math test results of students of class VII. Draw a Cumulative frequency table for the given data.

Marks	50	60	70	80	90
Number of students	5	8	12	6	4

Solution:

We total the frequencies as we proceed down the table to determine the cumulative frequency. This is how it is calculated:

Score	Frequency	Cumulative Frequency
50	5	5
60	8	13
70	12	25
80	6	31
90	4	35

Example 2: A group of students’ results on a science test are displayed in the table below. Find the cumulative frequency.

Score	Frequency
60	4
70	7
80	9
90	6
100	4

Solution:

Determine the cumulative frequency:

Score	Frequency	Cumulative Frequency
60	4	4
70	7	11
80	9	20
90	6	26
100	4	30

Example 3: In a group setting, students were asked how many hours they spent on homework each week. Determine the more than cumulative frequency of the given data.

Hours	Frequency
1-5	7
5-10	12
10-15	9
15-20	5
20-25	3

Solution:

We begin by computing the cumulative frequency in order to determine the more than cumulative frequency.

First, determine the cumulative frequency:

Hours	Frequency	Cumulative Frequency
1-5	7	7
5-10	12	19
10-15	9	28
15-20	5	33
20-25	3	36

Now, Determine the frequency that is more than cumulative:

Hours (Class Interval)	Hours	Frequency	More than Cumulative Frequency
1-5	More than 1	7	36
5-10	More than 5	12	36-7 = 29
10-15	More than 10	9	29-12 = 17
15-20	More than 15	5	17-9 = 8
20-25	More than 20	3	8-5 = 3

Cumulative Frequency Practice Problems

Some practice problems on cumulative frequeny are,

Problem 1: Determine the cumulative frequency of the results of a group of students’ physics tests are shown in the table below:

Score	60	70	80	90	100
Frequency	4	8	10	6	2

Problem 2: Given a population’s ages and their cumulative frequency distribution:

Age	20-29	30-39	40-49	50-59	60-69
Cumulative Frequency	15	30	45	60	75

Calculate the total number of individuals in the population.

Problem 3: The hours that each student in the group studied are shown in the table below. Determine the frequency cumulatively.

Hours	1-5	6-10	11-15	16-20	21-25
Frequency	6	8	10	5	3

Problem 4: The following shows the weight of a group of people’s cumulative frequency distribution. Calculate the weighted median.

Weight	50-59	60-69	70-79	80-89	90-99
Cumulative Frequency	8	15	23	30	36

Problem 5: A group of students timed how long it took them to finish an exam in minutes. Here is the cumulative frequency:

Time(min)	0-5	6-10	11-15	16-20	21-25
Cumulative Frequency	10	22	30	40	45

Determine how many students participated in the quiz overall.

FAQs on Cumulative Frequency

What is Normal and Cumulative Frequency?

Normal frequency is the number of times an event occurs and cumulative frequency is the sum of all previous frequencies up to the required point.

What is Greater than Cumulative Frequency?

Greater than frequency is obtained by determining the cumulative total frequencies starting from the last class to the first class. Here, the cumulate begins from the highest to the lowest class.

What is meant by Cumulative Frequency?

Cumulative frequency is the sum total of all the previous frequencies up to a certain point.

How do you Find CF in Grouped Data?

We can easily find the cumulative frequency of a grouped data by first finding its regular frequency and than adding all the preceding frequencies.

What is Formula for Cumulative Frequency?

The formula of Cumulative frequency is given by [Tex]CF_i = \sum_{j=1}^{i} f_j [/Tex]

What is Cumulative also Known as?

Cumulative frequency is also known as ogive.

Distinguishes Between Cumulative Frequency and Regular Frequency?

Regular frequency is the number of times a certain data point appears, whereas cumulative frequency is the running total of frequencies up to a given point.