Recover histogram frequencies and missing bar heights
Multiply density by class width · Recover a missing density from a class count · Combine histogram counts before calculating shares
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Frequency-density histograms
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Multiply density by class width
- Recover a missing density from a class count
- Combine histogram counts before calculating shares
Reverse the density calculation
Frequency=class width×frequency density. A bar spanning 20 to 40 with density 0.4 has frequency 20×0.4=8. The area is measured using the axis scales, not in screen pixels or centimetres. Width 10 and density 0.4 gives frequency 4, even though it has the same height.
Use totals from the areas
The illustrated counts are 4+8+8=20. The final class share is 8/20=40%. When a question gives a class frequency and asks for a missing height, divide frequency by its width. A class of width 15 and frequency 12 needs density 0.8. Check the vertical axis really is density before using these rules.
Sources for this lesson
Pause and explain
For width 20 and density 0.4, is frequency 0.4?
Worked example
Read the final class frequency from the illustrated histogram.
Show the worked solution
- Width=40−20=20.
- Frequency=20×0.4.
Answer 8.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A histogram bar spans 5≤x<15 and has frequency density 1.2. Find its frequency.
Give me a hint
Use width 15−5.
Compare my reasoning
- Width=10.
- Frequency=10×1.2=12.
12 observations.
Look for these in your work
- Read both horizontal bounds
- Multiply width by density
A class has width 15 and frequency 12. Find the required density height.
Give me a hint
Divide frequency by width.
Compare my reasoning
- Density=12/15.
- Simplify to 0.8.
0.8.
Look for these in your work
- Reverse the area relation
- Use the class width
Bars over 1≤t<6 and 6≤t<16 have densities 2 and 1.5. Find the total frequency and the second-class share.
Give me a hint
Calculate each area first.
Compare my reasoning
- Counts are 5×2=10 and 10×1.5=15.
- Total=25 and second share=15/25.
25 observations; second-class share 60%.
Look for these in your work
- Convert heights to frequencies
- Use the combined count as denominator
Common mistakes
- Do not use raw screen dimensions as axis units.
- Read the vertical-axis label before interpreting area.
What can you explain now?
For width 20 and density 0.4, is frequency 0.4?
Compare with the explanation
No; frequency is 20×0.4=8.
Height is density, and area in axis units is frequency.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
For width 20 and density 0.4, is frequency 0.4?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.