Estimate median, quartiles and interquartile range from a cumulative curve
Use cumulative counts N/4, N/2 and 3N/4 · Read the corresponding horizontal values · Subtract quartile values rather than count positions
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Cumulative frequency curves
- Linear interpolation
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
- Use cumulative counts N/4, N/2 and 3N/4
- Read the corresponding horizontal values
- Subtract quartile values rather than count positions
Count positions first, values second
For this cumulative-curve method with N=20, use counts 5,10,15 for Q1, median and Q3. Move horizontally from each count to the curve, then vertically to the value axis. The count 10 identifies where to read; it does not mean the median time is 10 minutes.
Estimate on the stated segments
Between cumulative counts 4 and 12, times run from 10 to 20. Q1≈10+(5−4)/8×10=11.25 and median≈10+(10−4)/8×10=17.5. Between counts 12 and 20, times run 20 to 40, so Q3≈20+(15−12)/8×20=27.5. IQR≈27.5−11.25=16.25 minutes.
These are estimates under the drawn interpolation model, not exact statistics of the hidden observations. For ordered raw lists, follow the quartile convention stated in the question; list methods and curve count-position methods need not give identical values for a small data set.
Sources for this lesson
Pause and explain
For N=20, does the count N/2=10 mean the median time must be 10 minutes?
Worked example
Estimate the median from the illustrated cumulative curve.
Show the worked solution
- Use cumulative count N/2=10.
- On the segment from (10,4) to (20,12), the fraction is (10−4)/8.
Answer Estimated 17.5 min.
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.
Use the illustrated straight-segment curve to estimate Q1.
Give me a hint
Read the time at cumulative count 5.
Compare my reasoning
- The class starts at time 10 with count 4.
- Move 1/8 of its width 10: 10+1.25.
Estimated Q1=11.25 min.
Look for these in your work
- Use N/4
- Interpolate within the correct class
Estimate Q3 and IQR from the same curve.
Give me a hint
Use count 15 and subtract the estimated Q1.
Compare my reasoning
- Q3=20+(15−12)/8×20=27.5.
- IQR=27.5−11.25=16.25.
Estimated Q3=27.5 min; IQR=16.25 min.
Look for these in your work
- Read the upper quartile value
- Use measurement units for the spread
A student subtracts count positions 15−5 and calls the IQR 10 minutes. Correct the result under the illustrated model.
Give me a hint
Count positions are not time measurements.
Compare my reasoning
- The time values are approximately 27.5 and 11.25.
- Their difference is 16.25 minutes.
Estimated IQR 16.25 min; 10 is a count-position difference.
Look for these in your work
- Separate the two axes
- Subtract the quartile values
Common mistakes
- A cumulative count is not the corresponding time value.
- IQR is a difference of data values, not a difference of count positions.
What can you explain now?
For N=20, does the count N/2=10 mean the median time must be 10 minutes?
Compare with the explanation
No; read the time where the curve reaches count 10.
The vertical count and horizontal measurement have different meanings.
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Make it stick
For N=20, does the count N/2=10 mean the median time must be 10 minutes?
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.