A histogram graph shows how data is distributed. It visually represents continuous, grouped data. It is different from a bar chart as it shows numerical data, while a bar chart measures categorical data. There are no gaps between the bars in a histogram. The area of each bar represents frequency rather than specifically the height. This means we have to calculate the frequency density before creating a histogram.
This article will explore what a histogram is and what separates it from a bar chart. We will explain the key elements of this graph and draw a histogram step-by-step. Histogram questions are included to test your knowledge and we explain common mistakes to avoid. This guide is aimed at GCSE Maths revision and is relevant for all major example boards, including Edexcel and OCR.
If you need further support, TeachTutti has qualified GCSE Maths tutors who can help you learn about and revise how to display data in a histogram.
What is a Histogram?
A histogram is a statistical graph. It shows continuous, grouped data. The graph looks similar to a bar chart. This can be unhelpful because it has key differences from a bar chart.
A bar chart:
- The frequency is represented by the height of the bar.
- There are gaps between the bars.
- It is used for discrete data, e.g. favourite colours.
A histogram:
- The frequency is represented by the area of the bar, not just its height.
- There are no gaps between the bars.
- It is used for continuous data, e.g. the heights of students in a class.
Frequency Density
Frequency is shown by the height of each bar in a normal bar chart. However, we can't do this just using height in a histogram. This is because the bars can have different widths. Instead, we use the frequency density, which allows us to compare data frequency, even when the class widths aren't equal. This is the formula: Frequency Density = Frequency / Class Width.
This is an example frequency table used to create a histogram:
| Class interval (x) | Frequency (f) | Class Width (w) | Frequency Density (f / w) |
| 0-5 | 10 | 5 | 2.0 |
| 5-15 | 20 | 10 | 2.0 |
| 15-25 | 15 | 10 | 1.5 |
| 25-40 | 30 | 15 | 2.0 |
| 40-60 | 25 | 20 | 1.25 |
It's important to understand what each axis represents:
- The x-axis - this is the horizontal line. It shows the continuous variable, such as time or height.
- The y-axis - this is the vertical line. It represents the frequency density (not the frequency).
We need to calculate the area of the bar to find its frequency, as the y-axis only shows the frequency density. This is the formula:
Frequency = Frequency Density × Class Width
Key elements of a Histogram
The structure of histograms follow specific rules to accurately represent data. To understand them properly, it's important to break the graph down into its key parts.
The Axes
There are two axes:
- The x-axis represents the continuous variable being measured. It is a continuous scale, rather than discrete categories. Examples include height, time and weight.
- The y-axis represents the frequency density. It's important to remember that it doesn't show the frequency, which is determined both by the height and the area of the bars.
Bars
- Bars must touch - Histograms represent continuous data. For this reason, there are no gaps between the bars.
- Varied bar widths – A Histogram differs from a bar chart in that the class widths can vary and bars can have different widths.
- The height of each bar represents the frequency density.
In the image above, the first graph is correct because the bars are touching and the frequency density is labelled. The second image shouldn't have gaps between the bars. It also shouldn't have equal widths when the class intervals are different.
Class widths
Class width means the range covered by each bar. To find the class width, subtract the lower boundary from the upper boundary of each class. This is the formula: Class width = Upper boundary - Lower boundary.
A histogram may have equal class widths. You must adjust the calculations to ensure the frequency density is correct if it contains unequal class widths. This is an example frequency table with the class width:
| Class interval | Lower boundary | Upper boundary | Class width | Frequency |
| 0-10 | 0 | 10 | 10 | 8 |
| 10-20 | 10 | 20 | 10 | 12 |
| 20-50 | 20 | 50 | 30 | 15 |
Frequency density
Frequency density must be used because the class width can vary. This guarantees that the bar area correctly represents the number of data points in that class.
You can find the frequency density with the following formula: Frequency density = Frequency / Class Width. It is important to memorise this formula - it is crucial when drawing or interpreting histograms.
For example, we have a class interval of 10-20 and a frequency of 15. We need to calculate the frequency density to show this in a graph using the formula above:
- Find the class width - The lower boundary is 10 and the upper boundary is 20. The width is 20-10 = 10.
- Divide the frequency by the width - 15/10 = 1.5.
- This means the bar height for the interval 10-20 is 1.5.
Use the area of bars to calculate frequency
The frequency of the class interval is shown by the area of each bar. When you have the frequency density, use the following formula to calculate the frequency: Frequency = Frequency Density x Class Width.
You can use this formula when approaching exam questions that require you to work backward from a given histogram to find missing frequencies.
How to Draw a Histogram
To draw a histogram accurately, you must understand frequency density and class widths. Below, we have outlined a step-by-step process for creating a histogram.
Step 1 - Frequency table
You will normally be given data in a grouped frequency table. This table will include:
- Class Intervals, such as 0–10, 10–20.
- Frequency - how many data points are in each class.
- Class Width - the difference between the upper and lower boundaries of the interval.
- Frequency Density - this is calculated as: Frequency Density = Frequency / Class Width.
| Class interval | Frequency | Class width | Frequency density |
| 0-10 | 5 | 10 | 0.5 |
| 10-30 | 12 | 20 | 0.6 |
| 30-50 | 8 | 20 | 0.4 |
Step 2 - Axes
Begin by drawing the axes for your graph:
- The x-axis is the horizontal line. It should be labelled with the class intervals. It must be on a continuous scale. For example, going up in increments of 10 (0, 10, 20, 30 etc).
- The y-axis is the vertical line. Label this line "Frequency Density".
Step 3: Draw the bars
The height of the bar is decided by its frequency density. The width of the bar is decided by its class width. If a class interval has a larger width, the bar will be wider but may not be taller. If a class has a higher frequency, the bar will have a greater area.
For example, let's say a class interval 0–10 has a frequency of 20, while a class interval 10–30 has a frequency of 30. The second bar may not be taller because the class width is also larger.
Step 4: Check for mistakes
There are several shortfalls to avoid when drawing a histogram. Make sure you do the following:
- Label the vertical axis as "Frequency Density" rather than "Frequency".
- If the class widths are unequal, the bars should not have the same width.
- There shouldn't be any gaps between the bars.
Interpreting Histograms
Unlike bar charts, the height of a bar doesn't represent frequency. Finding the frequency from a histogram requires more interpretation.
Frequency
The vertical axis represents frequency density. To find the frequency of a class interval, we need to use this formula: Frequency = Frequency Density x Class Width.
Follow these steps to find the values that fall within a range:
- Find the frequency density (the height of the bar).
- Identify the class width (the width of the bar).
- Multiply both values to calculate the frequency.
Comparing data
Histograms allow us to compare data distributions, including the following:
- A taller bar means a higher frequency density. It doesn't mean there are more values in this grouping.
- A wider bar has a larger class width. This can be calculated with the frequency density (the y-axis) to find the frequency.
- A cluster of high bars means data is concentrated in this area on the graph.
For instance, let's say a graph shows the height of students in a school. A tall and wide bar in the middle shows that most students fall inside this height range.
Trends and distribution shapes
The shape of a histogram shows us how data is distributed. The patterns below are useful in statistics. They also help real-world data analysis e.g. understanding exam scores of population demographics:
- Symmetrical Distribution - The data is evenly spread. It often appears in a bell shape, showing normal distribution.
- Skewed Right/Positive Right Skew - Most data is concentrated on the left. There is a longer tail on the right.
- Skewed Left/Negative Skew - Most data is concentrated on the right. There is a longer tail on the left.
Common misconceptions
Reading histograms can lead to common mistakes considering their similarity with bar charts. These include the following:
- The tallest bar shows the highest frequency density. It doesn't show the highest frequency. To find this, you also need to consider the class width.
- The bars may not have the same width. Often, the class intervals will be different. Make sure to check the scale.
- There will not be gaps between the bars. Histograms display continuous data, so the bars will always be touching.
Common mistakes and misconceptions
There are several key errors to avoid when answering exam questions related to histograms. We have covered the most common mistakes with advice on how to avoid these pitfalls.
Confusing frequency with frequency density
The height of the bar represents frequency density. This is different to a bar chart, where we only need to consider the height of the bar. Instead, we find frequency by multiplying frequency density and class width.
Histograms vs bar charts
Histograms look like bar charts. However, bar charts represent discrete data, while histograms represent continuous data. This means there are no gaps in a histogram, unlike bar charts, Further, the y-axis on a bar chart is frequency, whereas this measures frequency in a histogram.
Class widths are not always equal
Class intervals often have unequal widths in these exam questions related to this type of graph. This means the bars may have different widths. You need to check class widths before preparing your answer. For example, a narrower bar may have a taller height but a lower frequency than a comparatively shorter bar with a greater class width.
Labelling axes incorrectly
Make sure you write "Frequency Density" for the vertical axis, rather than "Frequency". For the x-axis, you need to use a continuous numerical scale. Don't use category labels for this axis. Correct labelling ensures the histogram is read correctly, avoiding a mark deduction in your examinations.
Use the frequency formula
You will normally be required to work backwards from a given histogram in an exam question. This will reveal the missing values. For example, you may be asked to find the frequency density or calculate the total frequency for a class interval. Avoid the temptation to simply read the values directly from a histogram. Instead, use the relevant formula in your workings.
Practice questions
Now that we have explored the theory of histograms, it is time to test your understanding. A selection of sample exam questions is listed below. Some of these are theory-based, while others are visual questions.
Example 1 - Theory questions
1
Is this statement correct: "the tallest bar in a histogram always represents the class with the highest frequency"?
2
Which formula correctly calculates frequency density?
3
A histogram bar has a class width of 5 and a frequency density of 3. What is the frequency for this class?
Example 2 - Visual questions
4
Which histogram is correct?
5
What shape is the above histogram?
6
Which class interval has the highest frequency?
Conclusion | Histogram, GCSE Maths
Unlike bar charts, histograms need careful attention to find the frequency of a data range, found using a formula that calculates frequency density and class widths. We have explored how it differs from a bar chart, the key elements in this graph and how to draw a histogram. There are common mistakes to avoid, most notably the belief that the highest bar has the greatest frequency.
The best approach to cementing your understanding of this graph is through practice. For further reading, MathGenie has a past paper on histograms. You can also create a histogram using the free tool by StatsKingdom.
If you need help getting to grips with this topic, TeachTutti has qualified GCSE Maths tutors who can support your revision. Each tutor has an enhanced DBS check. Lessons can be online or in-person.
Frequently asked questions
A histogram represents numerical data. The bars touch because the data is continuous, while the area of each bar represents frequency. A bar chart shows discrete or categorical data. The bars have gaps between them - the height of the bar represents frequency.
The class intervals in this graph can have unequal widths. If the frequency is plotted directly on the y-axis, the bars with wider class intervals will look larger. This would be misleading. Frequency density ensures that the area of the bar correctly shows the frequency. The formula is: Frequency Density = Frequency / Class Width.
Use this formula to calculate the area of the corresponding bar: Frequency = Frequency Density × Class Width. For instance, a bar has a frequency density of 2.5 and a class width of 4. The frequency is 2.5 × 4 = 10.
Histograms represent continuous data - the values flow uninterrupted between each other. The bars touch to show the continuity of the data, as there is no separation between groups.
- A symmetrical histogram suggests a normal distribution of data. This is usually bell-shaped.
- A right or positively skewed graph has a longer tail on the right. This means that most of the data is concentrated on the left.
- A left or negatively skewed histogram has a longer tail on the left. This means that most of the data is concentrated on the right of the graph.
You should never assume this. The class widths for each bar can vary. To guarantee the area of each bar correctly shows the frequency, you need to use frequency density.
This post was updated on 12 Mar, 2025.
