Frequency Distribution – An Essential Concept in Statistics

UNIT – 2
Frequency
Distribution
By: Prof. Priya Singh,
Assistant Professor School
of Commerce and
Management

Frequency Distribution
A frequency distribution shows the frequency of repeated items in a graphical form
or tabular form. It gives a visual display of the frequency of items or shows the
number of times they occurred. Let's learn about frequency distribution in this
article in detail.
What is Frequency Distribution?
Frequency distribution is used to organize the collected data in table form. The data
could be marks scored by students, temperatures of different towns, points scored in a
volleyball match, etc. After data collection, we have to show data in a meaningful
manner for better understanding. Organize the data in such a way that all its features
are summarized in a table. This is known as frequency distribution.
Let's consider an example to understand this better. The following are the scores of 10
students in the G.K. competition released by Mr. Ram 15, 17, 20, 15, 20, 17, 17, 14,
14, 20. Let's represent this data in frequency distribution and find out the number of
students who got the same marks.

Types of Frequency Distribution
There are four types of frequency distribution under statistics which are explained
below:
•Ungrouped frequency distribution: It shows the frequency of an item in each
separate data value rather than groups of data values.
•Grouped frequency distribution: In this type, the data is arranged and separated
into groups called class intervals. The frequency of data belonging to each class
interval is noted in a frequency distribution table. The grouped frequency table
shows the distribution of frequencies in class intervals.
•Relative frequency distribution: It tells the proportion of the total number of
observations associated with each category.
•Cumulative frequency distribution: It is the sum of the first frequency and all
frequencies below it in a frequency distribution. You have to add a value with the
next value then add the sum with the next value again and so on till the last. The last
cumulative frequency will be the total sum of all frequencies.

Frequency Distribution Table
• A frequency distribution table is a chart that shows the frequency of
each of the items in a data set. Let's consider an example to understand
how to make a frequency distribution table using tally marks. A jar
containing beads of different colors- red, green, blue, black, red, green,
blue, yellow, red, red, green, green, green, yellow, red, green, yellow.
To know the exact number of beads of each particular color, we need
to classify the beads into categories. An easy way to find the number
of beads of each color is to use tally marks. Pick the beads one by one
and enter the tally marks in the respective row and column. Then,
indicate the frequency for each item in the table.

Ungrouped Frequency Distribution Table:
In the ungrouped frequency distribution table, we don't make class intervals, we write the accurate
frequency of individual data. Considering the above example, the ungrouped table will be like this.
Given below table shows two columns: one is of marks obtained in the test and the second is of
frequency (no. of students).
Marks obtained in
Test
No. of Students
5 3
10 4
15 5
18 4
20 4
Total 20

Important Notes:
• Following are the important points related to frequency distribution.
• Figures or numbers collected for some definite purpose is called data.
• Frequency is the value in numbers that shows how often a particular item occurs in the given data
set.
• There are two types of frequency table - Grouped Frequency Distribution and Ungrouped
Frequency Distribution.
• Data can be shown using graphs like histograms, bar graphs.

Raw Data
• Raw data is the unorganized data when we’re done with the collection stage. This is because it is
similar to a lump of clay with no identity and also of no practical use. Definitely, we need to
organize this raw data. It is important to realize that organized data facilitates comparison and
meaningful conclusions.
• Further, to organize the data we need to look for similarities or group the data. In this way, we
effectively convert heterogeneous data into homogeneous data. To do so, an investigator has to
classify the data in the form of a series.
Series refer to those data which are in some order and sequence. Thus, if we arrange the data in the
example mentioned in the introduction according to the classes in your school, we will eventually
classify the data in form of a statistical series. Note that we can also arrange them according to
their heights. Hence, this basis of the arrangement of raw data can vary from purpose to purpose.

Variable
A variable is simply something that can vary with time and we can measure this variation. In other words,
a variable is a characteristic or a phenomenon which is capable of being measured and changes its value
over time. A variable is classified into two:
1] Discrete
A discrete variable’s value changes only in complete numbers or increases in jumps. Thus the phenomenon or
characteristic, a discrete variable represents should be such that its value cannot be infractions but only in whole
numbers. For example, the number of children in a family can be 2, 3, 4 etc but not 2.5, 3.5 etc.
2] Continuous
A continuous variable assumes fractional values or its value does not increase in jumps. For example, the heights of
students, the weights of students and so on.

Classification of Data
• The main objective of the organization of data is to arrange the data in such a form that it becomes fairly easy to
compare and analyze. Generally, we can do this by distributing data into various classes on the basis of some
attribute or characteristic. This distribution of data into classes is the classification of data. Further, each division
of data is a class. All in all, through the process of classification we can group and divide data into classes
according to a general attribute, which facilitates comparison and analysis.
Objectives of Classification
• Simplification and Briefness: Classification presents data in a brief manner. Hence, it becomes fairly easy to
analyze the data.
• Utility: As classification highlights the similarity in the data, it brings out its utility.
• Distinctiveness: With the help of grouping data into different classes, classification also brings out the
distinctiveness in data.
• Comparability: As already mentioned, it facilitates comparison of data.
• Scientific Arrangement: Classification arranges data on scientific lines. Thus it also increases the reliability of
data.
• Attractive and Effective: Lastly, through the process of classification, data becomes effective and attractive.

Characteristics of a Good Classification
• Comprehensiveness: Classification should cover all the items of the data. In other words,
it should be so comprehensive that it classifies all items in some group or class.
• Clarity: There should be no confusion of the placement of any data item in a group or
class. That is, classification should be absolutely clear.
• Homogeneity: The items within a specific group or class should be similar to each other.
• Suitability: The attribute or characteristic according to which classification is done should
agree with the purpose of classification.
• Stability: A particular kind of investigation should be effected on the same set of
classification.
• Elastic: As the purpose of classification changes, one should be able to change the basis of
classification.

Marks obtained in quiz Number of
students(Frequency)
12 1
15 4
16 1
17 1
19 1
20 2
21 3
23 2
24 1
29 1
30 3
Total 20
Frequency Distribution Table (Ungrouped)

Practice Questions
1. The heights of 50 students, measured to the nearest centimetres, have been found to be as follows:
• 161, 150, 154, 165, 168, 161, 154, 162, 150, 151, 162, 164, 171, 165, 158, 154, 156, 172, 160, 170, 153, 159,
161, 170, 162, 165, 166, 168, 165, 164, 154, 152, 153, 156, 158, 162, 160, 161, 173, 166, 161, 159, 162, 167,
168, 159, 158, 153, 154, 159
• (i) Represent the data given above by a grouped frequency distribution table, taking the class intervals as 160
– 165, 165 – 170, etc.
• Calculate Class Mid Value, Findout the student who have highest height among all.
2. The scores (out of 100) obtained by 33 students in a mathematics test are as follows:
69, 48, 84, 58, 48, 73, 83, 48, 66, 58, 84, 66, 64, 71, 64, 66, 69, 66, 83, 66, 69, 71 81, 71, 73, 69, 66,
66, 64, 58, 64, 69, 69
Represent this data in the form of a frequency distribution.

3. The following are the marks (out of 100) of 60 students in mathematics.
16, 13, 5, 80, 86, 7, 51, 48, 24, 56, 70, 19, 61, 17, 16, 36, 34, 42, 34, 35, 72, 55, 75, 31, 52, 28,72,
97, 74, 45, 62, 68, 86, 35, 85, 36, 81, 75, 55, 26, 95, 31, 7, 78, 92, 62, 52, 56, 15, 63,25, 36, 54, 44,
47, 27, 72, 17, 4, 30.
Construct a grouped frequency distribution table with width 10 of each class starting from 0 – 9.
4. Example 2: 100 schools decided to plant 100 tree saplings in their gardens on world environment
day. Represent the given data in the form of frequency distribution and find the number of schools
that are able to plant 50% of the plants or more?
95, 67, 28, 32, 65, 65, 69, 33, 98, 96, 76, 42, 32, 38, 42, 40, 40, 69, 95, 92, 75, 83, 76, 83, 85, 62, 37,
65, 63, 42, 89, 65, 73, 81, 49, 52, 64, 76, 83, 92, 93, 68, 52, 79, 81, 83, 59, 82, 75, 82, 86, 90, 44, 62,
31, 36, 38, 42, 39, 83, 87, 56, 58, 23, 35, 76, 83, 85, 30, 68, 69, 83, 86, 43, 45, 39, 83, 75, 66, 83, 92,
75, 89, 66, 91, 27, 88, 89, 93, 42, 53, 69, 90, 55, 66, 49, 52, 83, 34, 36

Ms. Radhika collected data of the weights of the students of her class and prepared the
following table:
A student is to be selected randomly from her class for some competition.
The chances of selection of the student is highest whose weight (in lbs) is in the
interval___________.
Weight (in lbs) 64 69
− 64 69
− 70 75
− 70 75
− 76 81
− 76 81
−
Number of students 88 1515 25

Constructing Frequency Distributions
• The following set of marks was obtained by 50 students in a
Mathematics exam:
25, 40, 35, 28, 45, 60, 55, 42, 38, 49, 52, 30, 29, 33, 27, 61, 62,
39, 41, 48, 53, 34, 47, 58, 50, 37, 43, 57, 59, 54, 46, 31, 32, 36,
51, 44, 56, 26, 63, 64, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14.
a) Construct a frequency distribution table with class
intervals of 10 marks each.
b) Determine the cumulative frequency for each class.

The age distribution of employees in a company is given:
• The age distribution of employees in a company is given:
23, 29, 35, 41, 47, 53, 24, 36, 44, 52, 60, 31, 42, 50, 58, 64, 30, 40, 46,
55, 61, 27, 37, 45, 51, 57, 26, 33, 48, 54, 32, 39, 43, 56, 28, 34, 49, 59,
62, 25.
a) Organize the data into a frequency table with class widths of 5
years each.
b) Identify the modal class and Calculate cumulative frequency.

Understanding Frequency & Intervals
3. What is the range of the data set: 5, 12, 18, 24, 31, 36, 42, 49, 53, 60?
4. The third class interval in a frequency distribution has a frequency of 12. What does this
mean in the context of the data?
5. A class interval is 30-40. Calculate its midpoint.
6. Given that the total observations are 100 and the frequency of the 50-60 class interval is 18,
find the percentage frequency of this class.
7. The lowest value in a dataset is 13, and the highest value is 89.
a) Determine the range.
b) If we create 8 classes, what should be the class width?

Cumulative & Relative Frequency
A dataset has the following class intervals and frequencies:
Class Interval Frequency
10 - 20 5
20 - 30 8
30 - 40 12
40 - 50 15
50 - 60 10
60 - 70 6

8.a) Construct the cumulative frequency distribution.
b) Find the relative frequency of the class 40-50.
9.A histogram is drawn for a dataset with class intervals of 5
units. What must be done if a class interval has a zero
frequency?
10.In a grouped frequency distribution, the class interval 25-
35 has a cumulative frequency of 22.
a) What does this cumulative frequency represent?
b) Find the cumulative frequency percentage for this
class if there are 50 total observations.

Class Interval (Defective Items) Frequency
0 - 10 3
10 - 20 7
20 - 30 12
30 - 40 9
40 - 50 5
Mean, Median, Mode in Frequency Distribution
•A company records the number of defective products produced in different shifts:
a) Find the modal class.
b) Determine the mean defective products using the midpoint method.

In a survey, people’s monthly electricity consumption (in kWh)
was recorded:
• 56, 72, 65, 80, 90, 100, 110, 120, 135, 145, 150, 170,
200
a) Find the median of this dataset.
b) Construct 5 equal class intervals and distribute
the data accordingly.

Applying Frequency Distributions
The weights (in kg) of students in a class are recorded:
35, 38, 42, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105,
110
a) Divide the data into 7 equal class intervals and prepare a
frequency table.
b) Find the class midpoint of the 50-60 interval.

A teacher collected students’ study hours per week:
• 4, 6, 7, 10, 12, 15, 20, 22, 24, 30, 35, 40, 45, 50, 55
a) Create a frequency distribution table with intervals of 10.
b) Determine the cumulative frequency of the class 20-30.

• The daily sales (in units) of a product for 30 days were recorded. Find the modal
sales interval and explain its significance.
13.The number of customers visiting a store per day was recorded over a month. If the
largest class interval (50-60 customers) has the highest frequency, what does this
indicate about the store’s footfall pattern?
14.A dataset of exam scores shows that the highest frequency is in the class interval
60-70. How would this affect decision-making for extra coaching sessions?
15.The number of employees in different salary brackets is given. If the cumulative
frequency of the 40-50k bracket is 85, interpret this data in terms of employee
earnings.
16.The monthly rainfall (in mm) was recorded for a year. Create a histogram and
explain how it helps visualize seasonal trends.
17.The blood pressure levels of patients in a hospital were categorized into class
intervals. If a majority fall in the 120-130 range, what health insights can be
derived?

• Graphical Representation is a way of analyzing numerical data. It exhibits the relation between data,
ideas, information, and concepts in a diagram. It is easy to understand and it is one of the most important
learning strategies. It always depends on the type of information in a particular domain. There are
different types of graphical representation. Some of them are as follows:
• Line Graphs – Line graph or linear graph is used to display continuous data and it is useful for predicting
future events over time.
• Bar Graphs – A Bar Graph is used to display the category of data and it compares the data using solid
bars to represent the quantities.
• Histograms – The graph that uses bars to represent the frequency of numerical data that are organized
into intervals. Since all the intervals are equal and continuous, all the bars have the same width.
• Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line
each time when that data occurs again.
• Frequency Table – The table shows the number of pieces of data that fall within the given interval.
• Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The
circle is considered with 100% and the categories occupied are represented with that specific percentage
like 15%, 56%, etc.
• Stem and Leaf Plot – In the stem and leaf plot, the data are organized from the least value to the greatest
value. The digits of the least place values from the leaves and the next place value digit from the stems.
• Box and Whisker Plot – The plot diagram summarizes the data by dividing it into four parts. The box
and whisker show the range (spread) and the middle ( median) of the data.

Frequency Distribution – An Essential Concept in Statistics

General Rules for Graphical Representation of Data
There are certain rules to effectively present the information in the graphical representation. They
are:
• Suitable Title: Make sure that the appropriate title is given to the graph which indicates the
subject of the presentation.
• Measurement Unit: Mention the measurement unit in the graph.
• Proper Scale: To represent the data in an accurate manner, choose a proper scale.
• Index: Index the appropriate colours, shades, lines, design in the graphs for better understanding.
• Data Sources: Include the source of information wherever it is necessary at the bottom of the
graph.
• Keep it Simple: Construct a graph in an easy way that everyone can understand.
• Neat: Choose the correct size, fonts, colours etc in such a way that the graph should be a visual
aid for the presentation of information.

Construct a frequency distribution table for the
following marks obtained by 30 students in a test:
45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 50, 55, 60, 65, 70, 75, 80,
85, 90, 95, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100
Represent the data using:
a) Histogram
b) Frequency Polygon

The number of customers visiting a shop in 10
days is recorded as follows:
•
120, 135, 150, 140, 160, 175, 130, 145, 155, 165
a) Prepare a frequency distribution table using class intervals of 10.
b) Draw a Bar Graph and an Ogive Curve.

Given the following distribution of student heights in cm,
construct a Histogram and a Cumulative Frequency
Curve:
Height (cm) Frequency
140 – 150 5
150 – 160 10
160 – 170 15
170 – 180 8
180 – 190 2

A company recorded the monthly salary
(in 1000s) of 50 employees as follows:
₹
Salary (₹) No. of Employees
10 – 20 5
20 – 30 8
30 – 40 12
40 – 50 15
50 – 60 10
Draw a Cumulative Frequency Polygon and find the median
salary.

a) Construct a frequency distribution table.
b) Represent the data using a Pie Chart and a Histogram.
Books Read No. of Students
0 – 5 6
5 – 10 12
10 – 15 18
15 – 20 9
20 – 25 5
A survey on the number of books read by 50 students in a semester gives the
following data:

The daily production of a factory (in metric tons) for
15 days is recorded as:
50, 55, 52, 58, 60, 65, 68, 70, 62, 57, 54, 59, 63, 66, 64
a) Form a grouped frequency distribution with class width 5.
b) Represent the data using a Histogram and a Frequency Polygon.

Frequency Distribution – An Essential Concept in Statistics

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