Use frequency density so bars of unequal width show data fairly; draw, read and estimate from histograms. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A histogram uses area for frequency. Frequency density =class widthfrequency. Bars touch. Unequal class widths make unequal bar widths; the tall bar is the one with high density, not necessarily high frequency. To read a frequency, compute density×width.
To draw one, add a frequency-density column to the table first. A 'how many between 20 and 35' question may split a bar. This is not a bar chart — labelling the vertical axis 'frequency' is the classic wrong axis.
Frequency density =class widthfrequency. The area of the bar is the frequency. Unequal widths make a frequency bar chart misleading — that is why density is used.
Objectives
Calculate frequency from frequency density and class width (Frequency = FD × Width).
Calculate frequency density from frequency and class width (FD = Frequency ÷ Width).
Combine bar areas to find total frequencies and compare intervals.
Key points
Calculate frequency density = frequency ÷ class width.
Draw a histogram with frequency density on the vertical axis and no gaps between bars.
Work backwards: frequency = frequency density × class width (the area of the bar).
Complete a table or a histogram when some information is missing.
Worked examples
Example 1
A histogram bar has frequency density 4 and class width 5. Calculate the frequency.
2 marks
State the histogram formula: Frequency = Frequency Density × Class Width.
Substitute given values: Frequency = 4 × 5.
Multiply: 4 × 5 = 20.
Answer: 20
Common slips
Slip: Add the two numbers together. Frequency is represented by the area of the bar.
Slip: Divide frequency density by class width: 4 ÷ 5. Area of a rectangle is width multiplied by height.
Slip: Write 9. 4 × 5 = 20.
Example 2
A class interval has frequency 30 and class width 10. Calculate the frequency density.
2 marks
State the frequency density formula: Frequency Density = Frequency ÷ Class Width.
Substitute given values: Frequency Density = 30 ÷ 10.
Calculate quotient: 30 ÷ 10 = 3.
Answer: 3
Common slips
Slip: Multiply frequency by class width: 30 × 10. Frequency density is the height of the bar, so divide area by width.
Slip: Divide width by frequency: 10 ÷ 30. Density is frequency per unit width.
Slip: Write 300. 30 ÷ 10 = 3.
Example 3
Calculate the total frequency of two bars, density 2 width 5 and density 4 width 10.
3 marks
Calculate frequency of the first bar: 2 × 5 = 10.
Calculate frequency of the second bar: 4 × 10 = 40.
Add the two bar frequencies: 10 + 40 = 50.
Answer: 50
Common slips
Slip: Add the densities together: 2 + 4 = 6. Each bar frequency must be calculated from its own area.
Slip: Add the widths together: 5 + 10 = 15. Multiply density by width for the second interval.
Slip: Write 50. Total frequency is 10 + 40 = 50.
Questions and answers
GCSE Maths histograms questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A histogram bar has frequency density 6 and class width 2. Calculate the frequency.
Show answer
Answer: 12
Area = density × width.
6 × 2 = 12.
M1 — Area = density × width.
A1 — 6 × 2 = 12.
Question 2 Easy2 marks
A histogram bar has frequency density 5 and class width 4. Calculate the frequency.
Show answer
Answer: 20
Area = density × width.
5 × 4 = 20.
M1 — Area = density × width.
A1 — 5 × 4 = 20.
Question 3 Medium2 marks
A class interval has frequency 18 and class width 6. Calculate the frequency density.
Show answer
Answer: 3
Density = frequency ÷ width.
18 ÷ 6 = 3.
M1 — Density = frequency ÷ width.
A1 — 18 ÷ 6 = 3.
Question 4 Medium2 marks
A class interval has frequency 40 and class width 5. Calculate the frequency density.
Show answer
Answer: 8
Density = frequency ÷ width.
40 ÷ 5 = 8.
M1 — Density = frequency ÷ width.
A1 — 40 ÷ 5 = 8.
Question 5 Hard2 marks
Which frequency is larger: density 6 width 4, or density 3 width 5?
Show answer
Answer: the first bar
First bar: 6 × 4 = 24. Second bar: 3 × 5 = 15.
24 > 15, so the first bar is larger.
M1 — First bar: 6 × 4 = 24. Second bar: 3 × 5 = 15.
A1 — 24 > 15, so the first bar is larger.
Question 6 Hard2 marks
Which frequency is larger: density 2 width 6, or density 5 width 4?
Show answer
Answer: the second bar
First bar: 2 × 6 = 12. Second bar: 5 × 4 = 20.
20 > 12, so the second bar is larger.
M1 — First bar: 2 × 6 = 12. Second bar: 5 × 4 = 20.
A1 — 20 > 12, so the second bar is larger.
Question 7 Exam-style4 marks
In a histogram, the bar for 20 < x ≤ 30 has height 4 and the bar for 30 < x ≤ 50 has height 2.5. The vertical axis is frequency density. Work out the total frequency for 20 < x ≤ 50.
Show answer
Answer: 90
The class 20 < x ≤ 30 has width 10, so its frequency is 4 × 10 = 40.
The class 30 < x ≤ 50 has width 20, so its frequency is 2.5 × 20 = 50.
Add the two frequencies: 40 + 50 = 90.
M1 — Class width 10 or 20 found from the boundaries
M1 — 4 × 10 or 2.5 × 20
A1 — 40 and 50
A1 — 90
Common mistakes
Most lost marks on this topic are one slip.
Adding density and width instead of multiplying
Frequency is the area of the bar (density × width)
Reading the vertical height as the frequency
The vertical axis is frequency density; multiply by width to get frequency
Comparing bar heights directly when widths differ
Always compare bar areas to determine which interval has more items
Spot the slip
A bar has frequency density 4 and class width 5. A student adds 4 and 5 to get 9. What is the frequency?
20
9
4
Show the slip
20 In a histogram, frequency is represented by the area of the bar: Frequency = frequency density × class width = 4 × 5 = 20.
Exam tips
A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Add 'width' and 'frequency density' columns to the table before drawing — the calculations are where the marks are.
Write 'frequency density' on the vertical axis label; an unlabelled axis can lose a mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.