Estimate a grouped mean using class midpoints
Calculate class midpoints · Weight midpoint values by frequencies · Explain why the result is an estimate
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
- Frequency-weighted means
- Continuous class intervals
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
- Calculate class midpoints
- Weight midpoint values by frequencies
- Explain why the result is an estimate
Replace unknown individual values by representatives
For classes 0≤t<10, 10≤t<20 and 20≤t<30 with frequencies 2,5,3, use midpoints 5,15,25. The estimated total is 2×5+5×15+3×25=160, over ten observations, giving estimated mean 16 minutes. The original times within the classes are unknown.
An estimate is not an exact reconstruction
Midpoint weighting represents each observation by its class center. It is useful when the within-class values are unavailable, but different actual data can have the same grouped frequencies and different exact means. An open-ended class needs additional information or an explicit representative-value assumption before this method can be applied.
Sources for this lesson
Pause and explain
Does a midpoint mean equal the exact mean of the hidden individual observations?
Worked example
Estimate the mean for the illustrated grouped table.
Show the worked solution
- Midpoints are 5,15,25 and the weighted total is 160.
- Total frequency is 10.
Answer Estimated mean 16 min.
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 the illustrated frequencies 2,5,3 in ten-minute classes from 0 to 30, estimate the mean.
Give me a hint
Use midpoints 5,15,25.
Compare my reasoning
- Weighted total=10+75+75=160.
- Divide by total frequency 10.
Estimated 16 min.
Look for these in your work
- Use class midpoints
- Retain frequency weights
Classes 0≤x<4 and 4≤x<10 have frequencies 3 and 2. Estimate the mean.
Give me a hint
Midpoints are 2 and 7, despite unequal widths.
Compare my reasoning
- Estimated total=3×2+2×7=20.
- Total frequency is five.
Estimated mean 4.
Look for these in your work
- Find each midpoint from its own boundaries
- Divide by the frequency total
Two values are known only to lie in 0≤x<10. Can the grouped midpoint estimate 5 be claimed as their exact mean? Give two compatible actual data sets.
Give me a hint
Compare values near the two ends of the interval.
Compare my reasoning
- The data 1,1 have mean 1.
- The data 9,9 have mean 9; both share the same grouped count.
No; (1,1) and (9,9) are counterexamples with means 1 and 9.
Look for these in your work
- Demonstrate the lost information
- Separate estimate from exact value
Common mistakes
- Weight midpoints by frequencies.
- State that the original measurements are unknown.
What can you explain now?
Does a midpoint mean equal the exact mean of the hidden individual observations?
Compare with the explanation
Not necessarily; it is an estimate.
Measurements can lie anywhere inside their stated classes.
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Make it stick
Does a midpoint mean equal the exact mean of the hidden individual observations?
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.