Grouping Data and Histogram:
Definitions, Properties, Examples
Dr. Pellumb Kllogjeri
Grouping Data and Histogram: When dealing with enormous amounts
of data, data grouping is quite essential. A pictogram or a bar graph can
alternatively be used to illustrate this data. Grouped data is data created
by individual grouping observations of a variable into groups. A
frequency distribution table of these groups may be used to summarise
or analyse the data.
A histogram is a graph that displays frequencies for intervals of values
of a metric variable. A histogram is similar to a bar graph. However, it is
used for class intervals that are not interrupted. It also displays a set of
continuous data’s underlying frequency distribution.
Grouping Data
When the number of observations is considerable, we can use the
grouping of data idea to separate the data into several categories.
Individual observations of a variable are grouped into groups, and the
frequency distribution table of these groups is a helpful way to
summarise the data.
Frequency Distribution
The data gathered is organised in a table using a frequency distribution.
Students’ grades, weather in other cities, match points, and so on might
all be incorporated in the data. Following the collection of data, we must
present it in a meaningful manner to aid comprehension. Arrange the
data in a table such that all of its feature s are summarised.
Let us consider an example. The following are the temperature
of 20 cities in March in degrees.
30,25,27,20,25,30,20,27,24,24,25,27,20,30,33,25,33,33,27,20
Let us present this data in a table and determine the frequency (the
number of repetitions of a value ) of the cities with the same
temperature.
Temperature
Number of Cities
20
4
24
2
25
4
27
4
30
3
33
3
All the collected data is arranged under the temperature column and the
number of cities columns, as can be seen. This arrangement makes it
simple to understand the information provided, and we can see the
number of cities with the same temperature.
Steps to Draw Frequency Distribution Table
for Grouped Data
We will follow the below steps to draw a frequency distribution table for
the grouped data.
1.
2.
3.
4.
Divide the data into groups using the information provided.
Sort the observations into ascending order.
Find out the frequency of each observation.
In the frequency distribution table, write the frequency and group
name.
Advantages of Grouping Data
Below are some of the advantages of grouping data.
1. Grouping data aids in focusing on crucial subpopulations while
ignoring others that aren’t.
2. Data grouping enhances estimating accuracy and efficiency.
Ungrouped Data
Raw data, also known as ungrouped data, is information collected from
direct observation.
20 students of a class in a
•
Example: Consider the marks of
particular exam.
•
40,50,50,56,92,60,70,60,88,76, 88,80,70,72,92,36,40,40,70,36
In the above example, the number of students who obtained the same
number of marks is called the frequency . Here, 3 students
got 40 marks. So, the frequency of 40 is 3.
Histogram
A histogram is a graphical depiction of a set of data that is divided into
user-defined ranges. Like a bar graph, the histogram turns a data series
into an easily understandable visual by grouping data points into logical
ranges or bins.
The histogram is made up of a series of bars (similar to a bar chart), but
these bars are adjacent to one another, and the height of the bars is
proportional to the frequency of the various classes . The area of each
rectangle denoted the frequency of each class.
The rectangles all have the same width, and their heights directly match
the class frequencies when the class intervals are equal.
If the length of the appropriate class interval rises, the height of a
rectangle must be proportionally reduced.
Histogram for a Grouped Data
A histogram is a two-dimensional graphical depiction of a continuous
frequency distribution.
The bars in a histogram are always put side by side, with no gaps
between them. That is, rectangles are built on the distribution’s class
intervals in histograms. The frequencies are proportional to the rectangle
areas.
Let us now examine the procedures involved in creating a histogram for
grouped data.
1. If the data is in a discontinuous form, represent it in a
continuous form.
2. On a uniform scale, mark the class intervals along the x−axis.
3. On a consistent scale, mark the frequencies along the y−axis.
4. Create rectangles with class intervals as the bases and
frequencies as the heights.
Histogram for Ungrouped Data
The histogram is created by plotting the class boundaries (not class
limits) on the x−axis and the corresponding frequencies on the y−axis
from the grouped data. Before constructing a histogram with ungrouped
data, we must first create a grouped frequency distribution.
Bar graphs are often used for discrete and categorical data. However, in
rare cases where an approximation is required, a histogram may be
generated. The steps for creating a histogram for ungrouped data are
as follows:
1. Mark the possible values on x−axis.
2. Mark the frequencies along the y−axis.
3. Draw a rectangle centred on each value, with equal width on
each side and a margin of 0.5 on either side.
Solved Examples on Grouping Data and
Histogram
Q.1. Draw a histogram for the below table, which represent the
marks obtained by 100 students in an examination:
Marks
Number of Students
0–10
5
10–20
10
20–30
15
30–40
20
40–50
25
50–60
12
60–70
8
70–80
5
Ans: The class intervals are all the same length, at ten marks each. Let’s
draw a line on the x−axis to represent these class intervals. Along
the y−axis, write the number of students on the appropriate scale. Below
is a representation of the histogram.
Scale: x−axis: 1cm=10 marks
y−axis: 1cm=5 students
The bars in the diagram above are drawn in a continuous pattern. The
rectangles have lengths (heights) that are proportionate to the
frequencies. The areas of the bars are proportional to the respective
frequencies because the class intervals are equal.
Q.2. Draw a histogram to represent the below data :
Class Interval
Frequency
0–10
8
10–20
12
20–30
6
30–40
14
40–50
10
50–60
5
Ans: The histogram for the given data is drawn below.
Scale: x−axis: 1cm=10 units, y−axis: 1cm=2 units
Q.3. A teacher wanted to analyse the performance of two sections of
students in a mathematics test of 100 marks. Looking at their
performances, she found that a few students got under 20 marks and
a few got 70 marks or above. So she decided to group them into
intervals of varying sizes: 0–20,20–30,….60–70,70–100. Then she
formed the following table:
Marks
Number of Students
0–20
7
20–30
10
30–40
10
40–50
20
50–60
20
60–70
15
70−above
8
Total
90
Ans: We need to make specific changes in the lengths of the rectangles
so that the areas are again proportional to the frequencies.
The steps to be followed are given below:
1. Select the class interval with the minimum class size.
2. The lengths of the rectangles are then modified to be proportionate to
the class size.
When the class size is 20, the length of the rectangle is 7. So, when the
class size is 10, the length of the rectangle will be (7/20)×10=3.5
Therefore, the modified table will be as follows.
Marks
Frequency
Width of the class
Length of the rectangle
0–20
7
20
(7/20)×10=3.5
20–30
10
10
(10/10)×10=10
30–40
10
10
(10/10)×10=10
40–50
20
10
(20/10)×10=20
50–60
20
10
(20/10)×10=20
60–70
15
10
(15/10)×10=15
70–100
8
30
(8/30)×10=2.67
Since we have calculated these lengths for an interval of 10 marks in
each case, we may call these lengths as “proportion of students
per 10 marks interval”.
So, the correct histogram with varying width is given in the below figure.
Q.4. In a study of covid patients in a village, the following
observations were noted. Represent the given data by using the
histogram.
Ages
Number of patients
10–20
3
20–30
6
30–40
13
40–50
20
50–60
10
60–70
5
Ans: The histogram for the given data is drawn below.
Scale: x−axis: 1cm=10 age
y−axis: 1cm=2 patients
Q.5. Draw the histogram for the below data.
Groups
Frequency
0–10
3
10–20
11
20–30
14
30–40
14
40–50
8
Ans: The histogram for the given data is drawn below.
Scale: x−axis: 1cm=10 units
y−y−axis: 1cm=5 units
Summary
In this unit, we learnt the definitions of grouping data, frequency
distribution, ungrouped data, and histograms. Also, we have studied the
method to draw histograms for grouped and ungrouped data and solved
some example problems on the same.
FAQs on Grouping Data and Histogram
Q.1. What do you mean by the grouping of data?
Ans: When the number of observations is considerable, we can use the
grouping of data idea to separate the data into several categories.
Individual observations of a variable are grouped into groups, and the
frequency distribution table of these groups is a helpful way to
summarise the data.
Q.2. What is grouping data and histogram?
Ans: When the number of observations is considerable, we can use the
grouping of data idea to separa te the data into several categories.
Individual observations of a variable are grouped into groups, and the
frequency distribution table of these groups is a helpful way to
summarise the data.
A histogram is a graphical depiction of a set of data that is d ivided into
user-defined ranges. Like a bar graph, the histogram turns a data series
into an easily understandable visual by grouping data points into logical
ranges or bins.
Q.3. What is the purpose of grouping data?
Ans: Grouping data aids in focusing on crucial subpopulations while
ignoring others that aren’t. Data grouping enhances estimating accuracy
and efficiency.
Q.4. How do you group data in statistics?
Ans: The following steps in the grouping can be summarised:
1. Determine the number of classes.
2. Calculate the range or the difference between the data’s highest and
lowest observations.
3. To estimate the size of the interval, divide the range by the number of
classes.
4. To calculate the upper-class limit, take the lower class limit of the
lowest class and multiply it by the class interval.
Q.5. What is an ungrouped data example?
Ans: Raw data, also known as ungrouped data, is information collected
from direct observation.
Example: Consider the marks of 20 students of a class in a particular
exam.
40,50,50,56,92,60,70,60,88,76,
88,80,70,72,92,36,40,40,70,36.
Homework
The heights in centimetres of 30 learners are given below.
142
163
169
132
139
140
152
168
139
150
161
132
162
172
146
152
150
132
157
133
141
170
156
155
169
138
142
160
164
168
Group the data into the following ranges and draw a histogram of the grouped data:
130≤h<140; 140≤h<150; 150≤h<160; 160≤h<170; 170≤h<180
(Note that the ranges do not overlap since each one starts where the previous one
ended.)