What is frequency density?
Think division.
Frequency density = frequency ÷ class width.
Histograms
Histograms display continuous data using bars with varying widths. They use frequency density to represent how data is distributed.
Overview
A histogram is used to display grouped continuous data.
The bars touch each other, and when class widths are unequal, the height of each bar represents frequency density, not frequency.
\( \text{Frequency Density} = \dfrac{\text{Frequency}}{\text{Class Width}} \)
This is the key idea in histograms. Wider class intervals must be adjusted, otherwise the graph would be misleading.
What you should understand after this topic
- Understand what makes a histogram different from a bar chart
- Understand why the bars in a histogram touch
- Calculate class width correctly
- Calculate frequency density
- Read and draw a histogram accurately
Key Definitions
Histogram
A graph for grouped continuous data.
Continuous Data
Data that can take any value in an interval.
Class Interval
A group of values, such as 10–20 or 20–30.
Class Width
The size of the interval.
Frequency
How many data values are in that class.
Frequency Density
The value used for the height of each bar in a histogram.
Key Rules
Bars touch
Because the data is continuous.
Use class width
Work out how wide each group is first.
Use frequency density
The bar height is usually not the raw frequency.
Area links to frequency
In a histogram, bar area represents the frequency.
Key Formulas
Frequency density
\( \text{Frequency Density} = \dfrac{\text{Frequency}}{\text{Class Width}} \)
Frequency
\( \text{Frequency} = \text{Frequency Density} \times \text{Class Width} \)
How to Solve
Step 1: Recognise a histogram
A histogram is used for grouped continuous data. The bars touch because the data is continuous.
Exam thinking: If class widths are different, you must use frequency density.
Histograms are usually drawn from grouped data in frequency tables.
Step 2: Understand area represents frequency
In a histogram, the area of each bar represents the frequency.
Key idea
Area = width × height
So frequency depends on both width and height.
Step 3: Find class width
The class width is the size of each interval.
For \(10 \le x < 20\), class width = \(20 - 10 = 10\).
Step 4: Calculate frequency density
Use the formula:
\( \text{Frequency Density} = \dfrac{\text{Frequency}}{\text{Class Width}} \)
Exam tip: Always calculate this before drawing the histogram.
Step 5: Draw the histogram
Common check: Width = class interval, height = frequency density.
- Draw the horizontal axis using class intervals.
- Label the vertical axis as frequency density.
- Draw each bar with correct width (class width).
- Set the height using frequency density.
- Ensure all bars touch.
Step 6: Read a histogram
You may need to find frequency from a histogram.
\( \text{Frequency} = \text{Frequency Density} \times \text{Class Width} \)
Exam thinking: Multiply height by width to get frequency.
Step 7: Reverse problems
Sometimes only the histogram is given, and you must find missing values.
Use the bar height as frequency density.
Multiply by class width to find frequency.
Complete any missing values in the table.
Step 8: Histogram vs bar chart
See bar charts for comparison.
Histogram
Continuous data, bars touch, area represents frequency.
Bar chart
Discrete data, bars do not touch.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on class width and frequency density.
Edexcel
Find the class width of \( 15 \leq x < 25 \).
Edexcel
A class interval has frequency 18 and class width 6.
Find the frequency density.
Edexcel
A histogram bar has class width 8 and frequency density 2.5.
Find the frequency represented by this bar.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on completing histogram tables.
AQA
The table shows information about the times some people spent in a shop.
| Time, \(t\) minutes | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 < t ≤ 10 | 20 | 10 | |
| 10 < t ≤ 30 | 70 | 20 | |
| 30 < t ≤ 35 | 22 | 5 | |
| 35 < t ≤ 50 | 30 | 15 | |
| 50 < t ≤ 60 | 8 | 10 |
Complete the frequency density column.
AQA
For the class interval \( 30 < t \leq 35 \), the frequency is 22.
Find the frequency density.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising interpreting histograms and understanding area.
OCR
A histogram bar has width 20 and frequency density 0.8.
Find the frequency represented by the bar.
OCR
One bar in a histogram is twice as wide as another bar.
Explain why you cannot compare the frequencies using only the heights of the bars.
OCR
Explain why the bars in a histogram touch.
Exam Checklist
Step 1
Work out the class width for each group.
Step 2
Calculate frequency density carefully.
Step 3
Label the vertical axis as frequency density.
Step 4
Make sure the bars touch and have correct widths.
Most common exam mistakes
Wrong axis
Writing frequency instead of frequency density.
Wrong width
Using the wrong class width in the calculation.
Height confusion
Thinking the tallest bar always has the highest frequency.
Bar gaps
Leaving spaces between bars like a bar chart.
Common Mistakes
These are common mistakes students make when drawing and interpreting histograms in GCSE Maths.
Using frequency instead of frequency density
Incorrect
A student labels the vertical axis as frequency.
Correct
Histograms use frequency density on the vertical axis, not frequency. Frequency density is calculated using \(\frac{\text{frequency}}{\text{class width}}\).
Forgetting to divide by class width
Incorrect
A student plots frequency directly without adjusting for class width.
Correct
You must divide the frequency by the class width to find frequency density before drawing the bars.
Drawing gaps between bars
Incorrect
A student leaves spaces between the bars.
Correct
Histograms represent continuous data, so the bars must touch with no gaps between them.
Using incorrect class width
Incorrect
A student calculates class width incorrectly from the intervals.
Correct
Class width is found by subtracting the lower boundary from the upper boundary. Use boundaries, not midpoints.
Assuming bar height equals frequency
Incorrect
A student compares bar heights directly as if they represent frequency.
Correct
In histograms, frequency is represented by the area of the bar, not just the height. Both height and width must be considered.
Try It Yourself
Practise interpreting and drawing histograms using frequency density.
Foundation
Foundation Practice
Understand frequency density and read simple histograms.
A class has frequency 20 and width 5. Find the frequency density.
Divide 20 ÷ 5.
20 ÷ 5 = 4.
A class has width 10 and frequency density 3. What is the frequency?
Multiply density × width.
3 × 10 = 30.
A class has width 4 and frequency density 6. Find frequency.
Multiply.
6 × 4 = 24.
What does the area of a bar in a histogram represent?
Think total data.
Area = frequency.
A class has frequency 15 and width 3. Find frequency density.
Divide.
15 ÷ 3 = 5.
Why are histograms used instead of bar charts?
Think data type.
Histograms show continuous data.
A class has width 2 and frequency density 8. Find frequency.
Multiply.
8 × 2 = 16.
A student uses frequency instead of density on the vertical axis. What is wrong?
Think vertical axis.
Histograms use frequency density.
A class has frequency 12 and width 6. Find frequency density.
Divide.
12 ÷ 6 = 2.
Higher
Higher Practice
Solve histogram problems involving frequency density and unequal class widths.
A class 0–5 has frequency 10. What is its frequency density?
Width = 5.
10 ÷ 5 = 2.
A class 10–20 has frequency density 4. Find frequency.
Width = 10.
4 × 10 = 40.
Which class has greater frequency if densities are equal?
Area matters.
Larger width → larger area → higher frequency.
A class 5–15 has frequency 30. Find frequency density.
Width = 10.
30 ÷ 10 = 3.
Why must unequal class widths use frequency density?
Think fairness.
Density ensures fair comparison.
A class 0–4 has density 5. Find frequency.
Width = 4.
5 × 4 = 20.
A student compares bar heights only. What is wrong?
Area matters.
Must consider width.
A class 20–30 has frequency 50. Find frequency density.
Width = 10.
50 ÷ 10 = 5.
What determines frequency in a histogram?
Think area.
Frequency = area.
A class 0–2 has density 6. Find frequency.
Width = 2.
6 × 2 = 12.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is frequency density?
Frequency divided by class width.
Why are bars different widths?
Class intervals can vary.
What is on the vertical axis?
Frequency density.