Find missing values and update a mean after a change
Use count×mean to recover the required total · Update both total and count when adding or removing an observation · Check a missing value by substitution
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Core and Extended foundations. 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
- Mean
- Simple linear equations
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
- Use count×mean to recover the required total
- Update both total and count when adding or removing an observation
- Check a missing value by substitution
A target mean fixes the total
Five values with mean 8 must total 40. If four are 3,7,9,11, their total is 30 and the missing value is 10. Verify (30+10)/5=8. Finding a mean is a division; finding the total reverses that division by multiplication.
What changed?
Four observations with mean 6 total 24. Adding a new value 9 gives total 33 and count 5, so new mean is 6.6. Replacing a value changes the total but keeps the count; removing one changes both. Increasing every value by the same amount increases the mean by that amount, while preserving the range.
Sources for this lesson
Pause and explain
When a fifth value is added to four observations, does the denominator stay four?
Worked example
Four values have mean 6. Add a fifth value 9 and find the new mean.
Show the worked solution
- Old total=4×6=24.
- New total=33 and count=5.
Answer 6.6.
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.
Five values have mean 8. Four are 3,7,9,11. Find the missing value.
Give me a hint
Recover the required total.
Compare my reasoning
- Required total=40; known total=30.
- Missing value=10.
10.
Look for these in your work
- Use count×mean
- Verify the resulting mean
Six values have mean 7. Remove a value of 12. Find the mean of the remaining observations.
Give me a hint
Update total and count.
Compare my reasoning
- Old total=42; remaining total=30.
- Five values remain, so mean=30/5=6.
6.
Look for these in your work
- Subtract the removed value
- Reduce the observation count
Four observations have mean 5. One recorded value 2 is corrected to 10. Find the corrected mean.
Give me a hint
This is a replacement rather than an added observation.
Compare my reasoning
- Old total=20; corrected total=20−2+10=28.
- Count stays four, so mean=7.
7.
Look for these in your work
- Keep the replacement count unchanged
- Correct only the affected contribution
Common mistakes
- Adding an observation changes the denominator.
- Replacing an observation keeps the count unchanged.
What can you explain now?
When a fifth value is added to four observations, does the denominator stay four?
Compare with the explanation
No; there are now five observations.
Both the total and the count must reflect the added observation.
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Make it stick
When a fifth value is added to four observations, does the denominator stay four?
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.