Calculate a mean using every observation and its frequency
Divide the observation total by their number · Use frequencies as weights · Explain why a discrete-data mean can be fractional
Section Paper 2 · Historical Extended · non-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
- Frequency distributions
- Fractions
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
- Divide the observation total by their number
- Use frequencies as weights
- Explain why a discrete-data mean can be fractional
Total divided by number of observations
The mean is the sum of all observed values divided by the number of observations. For 2,3,3,8 the total is 16 and there are four values, giving mean 4. A repeated observation contributes repeatedly to the total. The mean need not equal a value that actually appears.
Frequency tables preserve the same calculation
For values x with frequencies f, add each x×f and divide by the frequency total. The book distribution 1,2,3,4 with frequencies 2,1,3,2 gives total 21 and eight observations, so mean 21/8=2.625 books. Although each count is an integer, its average can be fractional. This is exact for the supplied discrete data, unlike a midpoint estimate for grouped measurements.
Sources for this lesson
Pause and explain
Must the mean of discrete counts be a whole number?
Worked example
Find the mean of 2,3,3,8.
Show the worked solution
- Sum=16 and count=4.
- Mean=16/4.
Answer 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.
Find the mean of 2,4,6,8.
Give me a hint
Add all four values.
Compare my reasoning
- Sum=20; count=4.
- 20/4=5.
5.
Look for these in your work
- Count every observation
- Divide by the count
Values 0,1,2 have frequencies 2,5,3. Find the mean.
Give me a hint
Use value×frequency.
Compare my reasoning
- Total=0×2+1×5+2×3=11.
- There are 10 observations.
1.1.
Look for these in your work
- Use frequency weights
- Keep a valid fractional mean
A student averages the distinct values 1,2,3,4 to get 2.5 for frequencies 2,1,3,2. Correct the mean and explain the difference.
Give me a hint
Restore repeated observations.
Compare my reasoning
- The weighted total is 21 and the count is 8.
- The distinct-value average wrongly gives each value the same weight.
21/8=2.625, using the eight observations.
Look for these in your work
- Preserve repetitions
- Explain the denominator
Common mistakes
- Divide by the number of observations, not distinct values.
- A fractional mean of integer counts can be valid.
What can you explain now?
Must the mean of discrete counts be a whole number?
Compare with the explanation
No; averaging integer observations can give a fraction.
The mean is a total shared across the number of observations.
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Make it stick
Must the mean of discrete counts be a whole number?
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.