Accumulate frequencies at the correct class boundaries
Add frequencies cumulatively · Plot cumulative counts at upper boundaries · Recover a class count by subtracting cumulative counts
Section Paper 2 · Historical Extended · non-calculator
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
- Grouped frequency tables
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
- Add frequencies cumulatively
- Plot cumulative counts at upper boundaries
- Recover a class count by subtracting cumulative counts
Running totals retain their boundary meaning
For classes 0≤t<10, 10≤t<20, 20≤t<40 with counts 4,8,8, cumulative counts below upper boundaries 10,20,40 are 4,12,20. Include the starting point (0,0). Plot at upper boundaries, not at class midpoints. Subtract adjacent cumulative counts to recover each class frequency.
Connecting points adds an estimation model
The illustrative curve uses straight segments between (0,0),(10,4),(20,12),(40,20), representing uniform distribution within each class for interpolation. The boundary counts are supplied, but intermediate counts are estimates. Cumulative counts never decrease. The point (0,0) marks a lower boundary with no smaller recorded values; it does not prove the minimum observation was exactly zero.
Sources for this lesson
Pause and explain
Should a cumulative count for below 20 be plotted at the class midpoint 15?
Worked example
Recover the class frequency for 10≤t<20 from the cumulative table.
Show the worked solution
- Below 20 there are 12 observations; below 10 there are 4.
- Subtract 12−4.
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.
For class counts 4,8,8 up to boundaries 10,20,40, give the cumulative counts.
Give me a hint
Add successive frequencies.
Compare my reasoning
- First count is 4.
- Then 4+8=12 and 12+8=20.
4,12,20.
Look for these in your work
- Accumulate rather than replace
- Match counts to upper boundaries
Upper-boundary cumulative counts are 2,7,10 at 10,20,30, with a starting count zero. Recover the three frequencies.
Give me a hint
Take successive differences.
Compare my reasoning
- First frequency=2.
- Second=7−2=5; third=10−7=3.
2,5,3.
Look for these in your work
- Subtract adjacent counts
- Check frequencies sum to the final count
A claimed cumulative table gives 4,12,11 at increasing boundaries 10,20,40. Can it represent observation counts?
Give me a hint
Cumulative counts cannot decrease.
Compare my reasoning
- The last class would have frequency 11−12=−1.
- A negative observation count is impossible.
No; the decrease implies a negative class frequency.
Look for these in your work
- Check monotonic cumulative totals
- Identify the impossible difference
Common mistakes
- Plot upper boundaries, not midpoints.
- A class-boundary start is not an observed minimum.
What can you explain now?
Should a cumulative count for below 20 be plotted at the class midpoint 15?
Compare with the explanation
No; plot it at the upper boundary 20.
The cumulative count describes all observations below that boundary.
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Make it stick
Should a cumulative count for below 20 be plotted at the class midpoint 15?
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.