Quartiles and interquartile range
Describe the middle half of an ordered distribution and state the convention used for a small list.
Section Paper 4 · Extended · Calculator
Extended only. 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 values and find medians.
- Subtract measures in the same units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does IQR measure?
What you will learn
- Interpret lower and upper quartiles.
- Calculate IQR and compare spread without relying on extremes.
Quartiles mark approximately one quarter and three quarters
Q1 is the lower quartile and Q3 the upper quartile; Q2 is the median. IQR = Q3 − Q1 measures the span of the middle fifty percent. For Q1 = 12 and Q3 = 25 minutes, IQR is thirteen minutes. It does not mean that fifty percent of observations equal thirteen.
For small discrete lists, quartile conventions can differ. Follow a method supplied in the question. In this lesson's list examples, use the median of the lower half for Q1 and median of the upper half for Q3; if the list has an odd size, exclude the overall median from the halves. Cumulative-frequency estimates use the N/4 and 3N/4 positions.
Use the stated convention consistently
For 2,4,6,8,10,12,14,16, the lower-half median is (4+6)/2 = 5 and upper-half median is (12+14)/2 = 13, giving IQR eight. The overall median is nine. These answers use the explicitly stated median-of-halves convention.
Range depends on both extremes, while IQR concentrates on the middle portion. A very large outlier can greatly increase range with little effect on IQR. When comparing distributions, describe typical value and spread together and keep any conclusion within the data's context.
Pause and explain
Q1 = 12, Q3 = 25. What is IQR?
Worked example
Using median-of-halves, find Q1, Q3 and IQR for 2,4,6,8,10,12,14,16.
Show the worked solution
- Lower-half median is (4+6)/2 = 5.
- Upper-half median is (12+14)/2 = 13.
- IQR = 13−5 = 8.
Answer Q1 = 5; Q3 = 13; IQR = 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.
Q1 = 8 kg and Q3 = 19 kg. Find IQR.
Give me a hint
Subtract using the same units.
Compare my reasoning
- IQR = 19−8.
- The result is eleven kilograms.
- It describes the span of the middle fifty percent.
11 kg.
Look for these in your work
- Subtracted quartiles.
- Kept measurement units.
Using median-of-halves, find Q1, Q3 and IQR for 1,3,5,7,9,11,13,15.
Give me a hint
Find the middle two values of each four-value half.
Compare my reasoning
- Q1 = (3+5)/2 = 4.
- Q3 = (11+13)/2 = 12.
- IQR = 12−4 = 8.
Q1 = 4; Q3 = 12; IQR = 8.
Look for these in your work
- Used the specified convention.
- Separated quartiles from the overall median.
Two groups have equal medians and IQRs 6 and 14. What limited comparison follows?
Give me a hint
IQR concerns only the central half.
Compare my reasoning
- Their reported median typical values are equal.
- The first has a narrower middle-half span.
- This alone does not compare extremes or prove one group performs better.
Equal medians; first group's middle half is less spread out.
Look for these in your work
- Compared central spread.
- Avoided unsupported claims about all data.
Common mistakes
- Subtracting Q3 from Q1.
- Switching quartile conventions.
- Treating IQR as the full observed range.
What can you explain now?
Why state a quartile method for a small list?
Compare with the explanation
Different accepted conventions can give different values.
Use the question's instruction consistently rather than silently switching methods.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, compare range and IQR and say which part of a distribution each describes.
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.