Combine group means using their observation counts
Recover a group total from its mean and size · Pool comparable observations with correct weights · Avoid an unweighted mean of unequal groups
Section Paper 3 · Historical Core · 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 as total divided by count
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
- Recover a group total from its mean and size
- Pool comparable observations with correct weights
- Avoid an unweighted mean of unequal groups
Recover totals before pooling
A group of two observations with mean 4 has total 8. A group of eight with mean 7 has total 56. Combined mean=(8+56)/(2+8)=6.4, not (4+7)/2=5.5. Each individual observation should receive the same weight in the pooled mean, so larger groups receive more total weight.
Check comparability and precision
Pool groups only when their variables and units are comparable and observations are not duplicated between groups. If supplied means were rounded, the recovered totals and pooled mean are estimates; do not claim greater accuracy than the source values justify. Exact means in these tasks permit exact totals.
Sources for this lesson
Pause and explain
Can you simply average two group means when the group sizes differ?
Worked example
Combine two values with mean 4 and eight with mean 7.
Show the worked solution
- Totals are 8 and 56.
- Divide total 64 by ten observations.
Answer 6.4.
Try a topic test
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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.
Three observations have mean 6 and seven have mean 10. Find the combined mean.
Give me a hint
Recover each total.
Compare my reasoning
- Totals=18 and 70.
- Combined mean=88/10.
8.8.
Look for these in your work
- Weight by counts
- Divide by the combined count
Five observations have mean 12 and five have mean 8. Find the combined mean and explain why an unweighted average works here.
Give me a hint
The two sizes are equal.
Compare my reasoning
- Combined total=60+40=100 over ten observations.
- Mean=10, also (12+8)/2 because the weights are equal.
10; equal group sizes justify equal weights here.
Look for these in your work
- Recognise the special equal-size case
- Still check totals
A group mean is reported only to the nearest whole unit. Can multiplying it by the group size recover the exact observation total?
Give me a hint
Rounding loses information.
Compare my reasoning
- The reported mean may differ from the true mean by up to half a unit, subject to the rounding convention.
- Multiplying cannot restore that lost information.
No; it gives an estimated total, not an exact reconstruction.
Look for these in your work
- Preserve the source precision
- Do not manufacture exactness
Common mistakes
- Do not give unequal groups equal weight.
- Keep units and observation membership consistent.
What can you explain now?
Can you simply average two group means when the group sizes differ?
Compare with the explanation
Not generally; weight by the observation counts.
Recovering the two totals preserves each individual observation’s contribution.
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Make it stick
Can you simply average two group means when the group sizes differ?
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.