Mean, median, mode and range
Choose and calculate summary measures for an ungrouped list and explain sensitivity to unusual values.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Order numbers.
- Add and divide accurately.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What must you do before finding a median?
What you will learn
- Calculate four common summary measures.
- Compare the effect of an extreme observation.
Different measures answer different questions
For scores 2, 4, 4, 6, 9, the mean is 25/5 = 5. The median is the middle ordered score four; the mode is the most frequent score four; the range is 9 − 2 = 7. Range measures spread, while the first three are averages describing a typical value.
Always sort before finding the median. With an even number of observations, take the mean of the two central values. A list can have more than one mode or no unique mode. Do not choose the largest value as the mode unless it is also most frequent.
An unusual value affects summaries differently
Replacing nine in the list with twenty-nine raises the mean to nine and the range to twenty-seven, but leaves the median and mode at four. The mean uses every observation, so it is sensitive to extremes; the median can better represent a skewed distribution.
Choosing an average depends on context. A mean can be useful for total-based planning, while a mode describes the most common category or discrete value. State the measure used and its units. An average by itself does not describe every observation or the entire distribution.
Pause and explain
What is the median of 1,3,5,9?
Worked example
Find mean, median, mode and range of 2,4,4,6,9.
Show the worked solution
- Total is twenty-five across five values, so mean is five.
- The ordered middle is four and the most frequent value is four.
- Range is nine minus two, giving seven.
Answer Mean 5; median 4; mode 4; range 7.
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.
Find the mean of 3,5,7,9.
Give me a hint
Divide the sum by the number of values.
Compare my reasoning
- The sum is twenty-four.
- There are four observations.
- Mean is 24/4 = 6.
6.
Look for these in your work
- Used the full total.
- Divided by the count.
Find median, mode and range of 8,2,5,2,9,4.
Give me a hint
Order the list first.
Compare my reasoning
- The ordered list is 2,2,4,5,8,9.
- Median is (4+5)/2 = 4.5; mode is two.
- Range is 9−2 = 7.
Median 4.5; mode 2; range 7.
Look for these in your work
- Used both central values.
- Separated frequency from spread.
The list 2,4,4,6,9 changes to 2,4,4,6,29. Which average remains four and why?
Give me a hint
Compare middle position and most frequent value.
Compare my reasoning
- The middle ordered value stays four.
- Four remains the most frequent value.
- Both median and mode remain four; the mean increases.
Median and mode remain 4.
Look for these in your work
- Explained each invariant.
- Recognised the mean's sensitivity.
Common mistakes
- Finding median before sorting.
- Dividing by the sum instead of the count.
- Confusing maximum with mode.
What can you explain now?
Is the range an average?
Compare with the explanation
No.
Range measures the span from minimum to maximum, not a typical central value.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, calculate the four summaries and justify one appropriate average in context.
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.