Histograms are a Higher-tier topic that appears regularly on AQA, Edexcel and OCR GCSE Maths papers. Unlike a bar chart, a histogram uses frequency density on the vertical axis and has no gaps between bars, because the data is continuous. The area of each bar — not its height — represents the frequency. This distinction catches many students out, so understanding it thoroughly is the key to unlocking full marks. This page explains frequency density clearly, walks you through drawing and interpreting histograms, provides worked examples, and finishes with exam-style practice questions. For context on related topics, see our complete GCSE Maths topics list.
What Is a Histogram?
A histogram displays the distribution of continuous data that has been grouped into class intervals. It looks similar to a bar chart, but there are crucial differences:
| Feature | Bar Chart | Histogram |
|---|---|---|
| Data type | Categorical or discrete | Continuous |
| Gaps between bars | Yes | No |
| Vertical axis | Frequency | Frequency density |
| What represents frequency | Height of bar | Area of bar |
| Class widths | Usually equal | Can be unequal |
Frequency Density
Frequency density adjusts for class intervals that may have different widths, ensuring a fair visual comparison.
Rearranging this gives:
This second formula is vital when you need to read frequencies from a histogram you have been given.
Why Use Frequency Density?
If class widths are unequal and you plot frequency on the vertical axis, wider classes automatically get taller bars even if they contain fewer data items per unit. Frequency density removes this bias by standardising the height relative to the class width.
Step-by-Step Method
Drawing a Histogram
- Calculate the class width for each interval. Be careful with boundaries — for example, 10 ≤ t < 20 has a class width of 10.
- Calculate the frequency density for each class: frequency ÷ class width.
- Draw the horizontal axis with a continuous scale (not category labels). Mark the class boundaries accurately.
- Draw the vertical axis labelled "Frequency density".
- Draw each bar from the lower to the upper class boundary, with the height equal to the frequency density. There must be no gaps.
Reading a Histogram
- Identify the class boundaries and class width for the bar you are interested in.
- Read the frequency density from the vertical axis.
- Calculate the frequency: frequency = frequency density × class width.
- If asked for a total, add the frequencies from all bars.
Estimating Frequencies Within a Class
Sometimes you need to estimate how many data items fall within part of a class. Because data is assumed to be evenly distributed within each class, calculate the proportion of the bar's area that corresponds to the required range.
Worked Example 1 — Higher Level
Question: The table shows the times taken by 60 runners to complete a 5K run.
| Time (min) | 15 ≤ t < 20 | 20 ≤ t < 25 | 25 ≤ t < 35 | 35 ≤ t < 50 |
|---|---|---|---|---|
| Frequency | 10 | 20 | 18 | 12 |
Draw a histogram for this data.
Working:
Calculate class widths and frequency densities:
| Class | Width | Frequency | Frequency density |
|---|---|---|---|
| 15 ≤ t < 20 | 5 | 10 | 10 ÷ 5 = 2.0 |
| 20 ≤ t < 25 | 5 | 20 | 20 ÷ 5 = 4.0 |
| 25 ≤ t < 35 | 10 | 18 | 18 ÷ 10 = 1.8 |
| 35 ≤ t < 50 | 15 | 12 | 12 ÷ 15 = 0.8 |
Draw the horizontal axis from 15 to 50 with a continuous scale. Draw bars with no gaps at the calculated heights: 2.0, 4.0, 1.8, 0.8. Label the vertical axis "Frequency density".
Worked Example 2 — Higher Level
Question: A histogram is drawn for the masses of parcels delivered to an office. The bar for the class 2 ≤ m < 5 has a frequency density of 6. How many parcels have masses in this class? Estimate how many parcels have masses between 2 kg and 3 kg.
Working:
Class width = 5 − 2 = 3.
Frequency = frequency density × class width = 6 × 3 = 18 parcels.
For the range 2 ≤ m < 3, the width is 1 out of a total class width of 3. Assuming even distribution:
Estimated frequency = 18 × (1 ÷ 3) = 6 parcels.
Common Mistakes
- Plotting frequency instead of frequency density — this is the single most common error. Always check whether class widths are equal; if they are not, you must use frequency density.
- Leaving gaps between bars — a histogram represents continuous data, so bars must touch.
- Incorrect class boundaries — for the interval "10 to under 20", the boundaries are 10 and 20, giving a width of 10, not 9.
- Misreading the vertical axis — if the examiner gives you a histogram, read the frequency density, not the frequency, from the axis.
- Forgetting to multiply back — when finding frequency from a histogram, remember frequency = frequency density × class width, not just the height.
Exam Tips
- Always show the frequency density calculation — even if the question only asks you to draw the histogram, writing the FD column earns method marks.
- Use a sharp pencil and ruler for accurate bars — examiners check boundary positions and heights.
- Label the vertical axis "Frequency density", not "Frequency". This small detail signals that you understand the concept.
- Check your total — add up all the frequencies (frequency density × class width for each bar) and verify it matches the total given in the question.
- Link to cumulative frequency — some Higher questions combine histograms with cumulative frequency graphs. See cumulative frequency and box plots for more.
- Revisit all key formulas on our GCSE Maths formulas page.
Practice Questions
Question 1: The table shows waiting times at a surgery.
| Time (min) | 0 ≤ t < 5 | 5 ≤ t < 10 | 10 ≤ t < 20 | 20 ≤ t < 40 |
|---|---|---|---|---|
| Frequency | 15 | 25 | 30 | 10 |
Calculate the frequency density for each class.
Question 2: On a histogram, a bar covers the interval 30 ≤ x < 45 and has a frequency density of 2.4. Find the frequency.
Question 3: A histogram shows the heights of 100 trees. The bar for 5 ≤ h < 10 has a frequency density of 8. Estimate how many trees are between 5 m and 7 m tall.
Question 4: Explain why a histogram is more appropriate than a bar chart for displaying continuous data with unequal class widths.
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Related Topics
- Frequency Tables and Grouped Data
- Cumulative Frequency and Box Plots
- Mean, Median, Mode and Range
- Sampling Methods
Summary
Histograms are a Higher-tier tool for displaying continuous grouped data. The vertical axis shows frequency density (frequency ÷ class width), and the area of each bar — not its height — represents the frequency. This makes histograms fair even when class widths are unequal. To draw one, calculate frequency densities, plot bars with no gaps against a continuous horizontal scale, and label axes correctly. To read one, multiply frequency density by class width to recover the frequency. Always show your working in a table, label the axis "Frequency density", and check that your total frequency matches the question. Mastering histograms links closely to cumulative frequency, box plots and estimating the mean from grouped data.