Histograms

When data is collected there are many ways that it can be displayed and organised, and numerical grouped data is often represented in a histogram. A histogram looks similar to a bar graph however as it represents grouped data it shows a range of variables. Since histograms are used to represent grouped data, the area of each bar on the histogram is proportional to the size of the class width which means that you can visually see if there are unequal group sizes. 

Histograms Histograms

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Contents
Table of contents

    How to plot a histogram

    When plotting a histogram, there is an x and y-axis. The x-axis represents the variable that is being measured and the y-axis represents the frequency density. The frequency density is the height of each bar on your histogram and can be calculated by dividing the frequency by the class width:

    frequency density = frequencyclass width

    All of the information needed to calculate the frequency density will be found in a table of data.

    The table represents the time taken in minutes for two PE classes of 30 students to complete a cross country course.

    Time, t (minutes)30≤t<4040 t<4545 t<5555 t<6060 t<65
    Frequency61618128

    As this is grouped data, you are able to plot it onto a histogram. The first step would be to find the frequency density for each class, and this can be done by creating a new table and completing a few calculations:

    Time, t (minutes)FrequencyClass WidthFrequency Density
    30 t<406100.6
    40 t<451653.2
    45≤ t<5518101.8
    55≤ t<601252.4
    60≤ t<65851.6

    The class width can be calculated by subtracting the lower boundary from the upper boundary, for example, 40-30=10.

    The frequency density is calculated by dividing the frequency by the class width, for example, 6÷10=0.6.

    Now that you have found the frequency density for each of the class boundaries, you can plot your histogram.

    Histograms Example of a histogram StudySmarterHistogram representing time taken to complete a cross country, Thomas-Gay - StudySmarter Originals

    When plotting a histogram, you need to use the data from the time, t, minutes column of the table and the frequency density column of the table.

    From this information on the histogram you are also able to create a frequency polygon. To do this you add a line from the middle of each bar on your histogram, and this is shown below.

    Histograms Frequency Polygon Example StudySmarterFrequency polygon example, Thomas-Gay - StudySmarter Originals

    Read more about Frequency Polygons.

    How to interpret a histogram

    When thinking about histograms it is not only important to be able to create your own, but also to understand and interpret one that you are given.

    The histogram below shows the test results as a percentage from a group of students' mathematics exam.

    Histograms Example of a histogram StudySmarterHistogram representing test results, Thomas-Gay - StudySmarter Originals

    Here are a set of questions that you can answer from the histogram above:

    1. How many students scored between 75% and 85% on the exam?

    2. How many students scored between 60% and 70% on the exam?

    3. How many students scored 80% or more on the exam?

    4. How many students took the exam in total?


    Now let's work through each question together!

    1. How many students scored between 75% and 85% on the exam?

    First, you start by looking at the histogram. You need to find the area of the bars between 75 and 85, and they are shaded below:

    Histograms Example of a histogram with one of the bars on the chart highlighted StudySmarterHistogram worked example, Thomas-Gay - StudySmarter Originals

    To find the shaded area we can simply multiply the height by the width:

    10×1=10

    Therefore we know that 10 students scored between 75% and 85%.

    2. How many students scored between 60% and 70% on the exam?

    This can be done in the same way, let's start by shading the area needed on the graph:

    Histograms Example of a histogram with one and a half of the bars on the chart highlighted StudySmarterHistogram worked example, Thomas-Gay - StudySmarter Originals

    For this area, you will need to break it down into two parts since you are looking at two different bars:

    5×1=5

    5×3=15

    5.5+15=20.5

    This shows you that around 20 students scored between 60% and 70% on the exam.

    3. How many students scored 80% or more on the exam?

    Once again let's start by shading the area that is needed, for this question you need to shade everything over 80%:

    Histograms Example of a histogram with one and a half of the bars on the chart highlighted StudySmarterHistogram worked example, Thomas-Gay - StudySmarter Originals

    This question again needs to be broken down into two parts:

    5×1=5

    15×0.6=9

    5+9=14

    This shows that 14 of the students scored 80% or more on the exam.

    4. How many students took the exam in total?

    For this question, you need to find the area of each bar, your shaded area will now look like this:

    Histograms Example of a histogram with all the bars on the chart highlighted StudySmarterHistogram worked example, Thomas-Gay - StudySmarter Originals

    You will need to find the area for each bar and add them all together to find the total number of students:

    5×1.2=610×0.4=410×1=105×3=155×3.2=1610×1=1015×0.6=9

    Now you can add them all together to find the total number of students.

    6+4+10+15+16+10+9=70

    Histograms - Key takeaways

    • A histogram is a type of graph that represents grouped data.

    • Frequency density is used to plot a histogram, and it is calculated by dividing the frequency by the class width.

    Frequently Asked Questions about Histograms

     What is a histogram?

    A histogram is a type of graph that represents grouped data.

    What is a histogram used for?

    A histogram is often used to visually show data, as it uses the frequency density, you are able to see how the variable ranges.

    How do you draw a histogram?

    A histogram can be drawn by calculating the frequency density, then plotting the frequency density and the class boundaries.

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