Calculate averages and measures of spread
Use mean, median, mode, quartiles, range and interquartile range for individual and frequency data, with grouped estimates at Extended level.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which average is the middle value after ordering the data?
What you will learn
- Find averages and spread from lists or discrete frequency tables
- Estimate a grouped mean and identify a modal class at Extended level
- Use a graphic display calculator for the specified data summaries
Choose a measure for the distribution
Mean uses every value; median is the central ordered value; mode is the most frequent value. Range = maximum − minimum; interquartile range = Q3 − Q1 and describes the middle half.
An outlier can pull the mean more than the median. Give units when the data have them and compare averages together with spread when describing two groups.
Frequency tables and grouped intervals differ
For a discrete frequency table, total frequency n = Σf and mean = Σfx/Σf. Cambridge permits a GDC for mean, median and quartiles for discrete data, and mean for grouped data.
For grouped data, use each class midpoint as an estimate and calculate Σ(f × midpoint)/Σf; identify the class with the greatest frequency as the modal class. The estimate is not an exact mean because individual values are unknown. Core lists and frequency tables are not grouped.
Pause and explain
For grouped data, what value represents each interval when estimating the mean?
Worked example
For data 2, 3, 3, 5, 7, 8, 10, find the mean, median, mode and range.
Show the worked solution
- Sum = 38 across 7 values, so mean = 38/7 ≈ 5.43.
- The middle (4th) ordered value is 5; the most frequent value is 3.
- Range = 10 − 2 = 8.
Answer Mean 38/7 (≈5.43), median 5, mode 3, range 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 values 2, 4 and 9, find the median and range.
Give me a hint
Order the values and subtract the minimum from the maximum.
Compare my reasoning
- The values are already ordered: 2, 4, 9.
- The middle value is 4.
- Range = 9 − 2 = 7.
Median 4; range 7.
Look for these in your work
- Values ordered
- Middle selected
- Range calculated
A frequency table has values x = 5, 10, 15 with frequencies 2, 3, 1. Find the mean and modal value.
Give me a hint
Use Σfx/Σf and find the value with greatest frequency.
Compare my reasoning
- Σf = 2 + 3 + 1 = 6.
- Σfx = 5×2 + 10×3 + 15×1 = 55.
- Mean = 55/6 ≈ 9.17; modal value is 10.
Mean 55/6 (≈9.17); mode 10.
Look for these in your work
- Frequencies totalled
- Weighted sum calculated
- Mode identified
Grouped travel times have intervals 0–10, 10–20, 20–30 minutes with frequencies 4, 8, 3. Estimate the mean and identify the modal class.
Give me a hint
Use midpoints 5, 15 and 25 and weight each by its interval frequency.
Compare my reasoning
- Estimated total = 5×4 + 15×8 + 25×3 = 215.
- Total frequency = 15, so estimated mean = 215/15 ≈ 14.3 minutes.
- The greatest frequency is 8, so 10–20 minutes is the modal class.
Estimated mean 14.3 minutes; modal class 10–20 minutes.
Look for these in your work
- Midpoints used
- Weighted estimate calculated
- Modal class selected
Common mistakes
- Using a class boundary instead of a midpoint for grouped estimates
- Forgetting to weight values by their frequencies
- Calling a grouped mean exact
- Comparing spread with different units or inconsistent methods
What can you explain now?
What is the range of 4, 7, 9, 12?
Compare with the explanation
8
Range = 12 − 4.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why is a mean from grouped intervals an estimate, and when might the median describe a data set better?
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.