Histograms

What is a Histogram?

A histogram is the graphical representation of data where data is grouped into continuous number ranges and each range corresponds to a vertical bar.

The horizontal axis displays the number range and the vertical axis (frequency) represents the amount of data that is present in each range.

The number ranges depend upon the data that is being used. 


Histogram Example

Uncle Bruno owns a garden with \(30\) black cherry trees. Each tree is of a different height. 

Histogram example of a cherry tree garden

The height of the trees (in inches) is given below:

61, 63, 64, 66, 68, 69, 71, 71.5, 72, 72.5, 73, 73.5, 74, 74.5, 76, 76.2, 76.5, 77, 77.5, 78, 78.5, 79, 79.2, 80, 81, 82, 83, 84, 85, 87

We can group the data as follows in a frequency distribution table by setting a range:

Height Range (ft) Number of Trees (Frequency)
60 - 75 3
66 - 70 3
71 - 75 8
76 - 80 10
81 - 85 5
86 - 90 1

This data can be now shown using a histogram.

In a histogram, there shouldn’t be any gaps between the bars.

histogram example


Histogram Calculator

Here is the Histogram Calculator.

Here, we can enter a list of values of data and it gives us the corresponding histogram.

This allows you to change the class interval also by sliding the slider.


Frequency Histogram

A frequency histogram is a histogram that shows the frequencies (the number of occurrences) of the given data items.

For example;

In a hospital, there are \(20\) newborn babies whose ages in increasing order are as follows:

1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5

This information can be shown in a frequency distribution table as follows:

Age (in days) Frequency
\(1\) \(4\)
\(2\) \(5\)
\(3\) \(8\)
\(4\) \(2\)
\(5\) \(1\)

This data can be now shown using a frequency histogram.

frequency histogram


Histogram Shapes

We have mainly \(6\) types of histogram shapes. They are:

  1. Bell Shaped Histogram
  2. Bimodal Histogram
  3. Skewed Right Histogram
  4. Skewed Left Histogram
  5. Uniform Histogram
  6. Random Histogram

Bell-Shaped Histogram

A bell-shaped histogram has a single peak.


 

bell shaped histogram shape

Example:

The following histogram shows the number of children visiting a park at different time intervals.

histogram example of a bell shaped histogram graph showing number of children visiting a park

The maximum number of children visit the park between 5.30 PM to 6 PM. 

The histogram has just one peak at this time interval and hence it is a bell-shaped histogram.

Bimodal Histogram

A bimodal histogram has two peaks and it looks as follows:

bimodal histogram shape

Example:

The following histogram shows the marks obtained by the \(48\) students of Class \(8\) of St.Mary’s School.

bimodal histogram shows the marks obtained by the 48 students of class 8

The maximum number of students have scored either between 40 to 50 marks OR between 60 to 70 marks.

This histogram has two peaks (between 40 to 50 and between 60 to 70) and hence it is a bimodal histogram.

Skewed Right Histogram

A skewed right histogram is a histogram that is skewed to the right. It looks as follows:

 

skewed right histogram shape

Example:

The following histogram shows the number of people corresponding to different wage ranges.

Skewed right histogram showing the number of people corresponding to different wage ranges

Here the bars of the histogram is skewed to the right. Hence, it is a skewed right histogram.

Skewed Left Histogram

A skewed left histogram is a histogram that is skewed to the left. It looks as follows:

 

skewed right histogram shape

Example:

The following histogram shows the number of students of Class \(10\) of Greenwood High School according to the amount of time they spent on their studies on a daily basis.

skewed left histogram showing the number of students of class 10 of Greenwood High School according to the amount of time they spent on their studies on a daily basis

Here the bars of the histogram is skewed to the left. Hence, it is a skewed left histogram.

Uniform Histogram

A uniform histogram is a histogram where all the bars are more or less of the same height. It looks as follows:

uniform histogram shape

Example:

Ma’am Lucy, the Principal of Little Lilly Playschool, wanted to record the heights of her students. The following histogram shows the number of students and their varying heights.

uniform histogram showing the heights of students

The height of the students ranges between 90 cms to 110 cms.

In this histogram, the lengths of all the bars are more or less the same. Hence, it is a uniform histogram.

Random Histogram

A random histogram is a histogram where the heights of bars have no pattern. It looks as follows:

random histogram shape

Example:

The following histogram shows the different age groups of the people residing in J P Nagar, Bangalore.

random histogram showing the different age groups of the people residing in J P Nagar, Bangalore

The residents belong to varying age groups. 

In this histogram, all bars are of different heights and there is no pattern. Hence, it is a random histogram.


Difference Between a Bar Chart and a Histogram

  • A bar graph is the graphical representation of categorical data using rectangular bars where the length of each bar is proportional to the value they represent. 
  • A histogram is the graphical representation of data where data is grouped into continuous number ranges and each range corresponds to a vertical bar.

The fundamental difference between histograms and bar graphs from a visual aspect is that bars in a bar graph are not adjacent to each other.

The main differences between a bar chart and a histogram are as follows:

Bar Graph Histogram
Equal space between every two consecutive bars. No space between two consecutive bars. They should be attached to each other.
X-axis can represent anything. X-axis should represent only continuous data that is in terms of numbers.

But in both graphs, Y-axis represents numbers only.

We can understand these differences from the following figure:


difference between bar graph and histogram


How to Calculate a Histogram?

The process of making a histogram using the given data is described using the following example:

Example:

Construct a histogram for the following frequency distribution table that describes the frequencies of weights of \(25\) students in a class.

Weights (in Kg) Frequency (Number of students)
30 - 35 4
35 - 40 10
40 - 45 8
45 - 50 4

Steps to draw a histogram:

Step 1:

We choose a suitable scale to represent weights on the horizontal axis.
Here, we can choose the scale to be \(1 \text{ cm} = 5 \text{ kg}\).
Since the weights in the table start from \(30\), not from \(0\), we give a break/kink on the X-axis.

Step 2:

We choose a suitable scale to represent the frequencies on the vertical axis.
Here we have frequencies varying from \(4\) to \(10\). Thus, we choose the scale to be \(1 \text{ cm} = 2 \).

Step 3:

We then draw the bars corresponding to each of the given weights using their frequencies.

Example of how to calculate a histogram.


Solved Examples

Example 1

 

 

The blood groups of \(20\) students of a class are recorded as follows:

A, B, O, O, B, A, O, O, O, AB, A, O, O, A, A, B, B, AB, O, B

Draw a frequency table to represent the given data and identify the rarest blood group among these \(20\) students.

Solution:

In the given data, the number of students with blood group :

  • \(A\)    = \(5\)
  • \(B\)    = \(5\)
  • \(O\)    = \(8\)
  • \(AB\) = \(2\)

The frequency distribution table of the data is:

Blood group Frequency
\(A\)  \(5\)
\(B\)  \(5\)
\(O\)  \(8\)
\(AB\)  \(2\)

From this table, the blood group with least frequency is \(AB\).

\( ∴\text{The rarest blood group = AB} \)
Example 2

 

 

Consider the following histogram that represents the weights of \(33\) newborn babies in a hospital.

A histogram example showingthe weights of 33 newborn babies in a hospital

If the children weighing between \(2\) kg to \(3\) kg are considered healthy, then find the percentage of the children of this hospital that are healthy.

Solution:

We have to first find the number of children weighing between \(2\) kg to \(3\) kg.

From the given histogram, the number of children weighing between:

  • \(2 \text { kg } 2.5 \text { kg }\) = \(10\)
  • \(2.5 \text { kg } 3 \text { kg }\) = \(18\)

Therefore, the number of children weighing between:

\[2 \text{ kg} \text { to } 3 \text{ kg} = (10+18=28)\]

The total number of children in the hospital = \(33\)

Hence, the required percentage is:
\[\frac{28}{33} \times 100 \approx 85\% \]

\( ∴\text{Required percentage} = 85\% \)
Example 3

 

 

A random survey is done on the number of children belonging to different age groups who play in government parks and the information is tabulated as follows:

Age (in years) Frequency
\(1 - 2\) \(8\)
\(2 - 4\) \(10\)
\(4 - 7\)  \(18\)
\(7 - 9 \) \(10\)
\(9 - 11 \) \(12\)
\(11 - 15 \) \(6\)

(i) Draw a histogram representing the data.

(ii) Identify the number of children belonging to the age groups \(2, 3, 4,5,6 \) and \(7\) who play in government parks.

Solution:

(i) We take the age (in years) on the horizontal axis of the graph and by observing the first column of the table, we choose the scale to be:

\[1 \text{ cm} = 1 \text{ year}\]

We take frequency on the vertical axis of the graph and by observing the second column of the table, we choose the scale to be:

\[1 \text{ cm} = 2\]

Now, we will draw the corresponding histogram.

A histogram example showing the data of a random survey done on number of children of different age group playing in parks

(ii) From the table/graph, the number of children belonging to the age groups:

  • \(2 \text { to } 4 \text { years }\) = \(10\)
  • \(4 \text { to } 7 \text { years }\) = \(18\)

So the number of children belonging to the age groups \(2, 3, 4,5,6 \) and \(7\) who play in government parks is

\[10+18= 28\]

\(∴\text{Required number of children } = 28 \)
 
important notes to remember
Important Notes
  1. A histogram is a graph that is used to summarise continuous data.
  2.  A histogram gives the visual interpretation of continuous data.
  3. The scales of both horizontal and vertical axes don’t need to start from \(0\).
  4. There should be no gaps between the bars of a histogram.

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

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Practice Questions

 
 
 
 
 
 

 


Frequently Asked Questions (FAQs)

1) What is a histogram used for?

A histogram is used for showing the frequencies of different data.

It is the graphical representation of data where the data is grouped into continuous number ranges and each range corresponds to a vertical bar.

2) What is a histogram graph?

A histogram is a type of graph for the graphical representation of data. This data is grouped into number ranges and each range corresponds to a vertical bar.

3) How do you construct a histogram?

The steps to construct a histogram are as follows:

Step 1:
We place the intervals on the horizontal axis by choosing a suitable scale. 

Step 2:
We place frequencies on the vertical axis by choosing a suitable scale.

Step 3:
We construct vertical bars according to the given frequencies.

4) What is the difference between a bar graph and a histogram?

The fundamental difference between histograms and bar graphs from a visual aspect is that bars in a bar graph are not adjacent to each other.

The main differences between a bar chart and a histogram are as follows:

Bar Graph Histogram
Equal space between every two consecutive bars. No space between two consecutive bars. They should be attached to each other.
X-axis can represent anything. X-axis should represent only continuous data that is in terms of numbers.
  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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