Read cumulative frequency diagrams
Build cumulative totals and estimate the median, quartiles, percentiles and interquartile range from an Extended cumulative frequency graph.
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
What is the final cumulative frequency?
What you will learn
- Create cumulative frequencies from grouped data
- Read median and quartiles at their cumulative positions
- Interpret estimates and recognise the information lost through grouping
Accumulate frequencies up to each boundary
Add class frequencies successively to make cumulative frequency. Plot each upper class boundary against the cumulative total and join the plotted points with a smooth increasing curve.
The final cumulative frequency is the sample size. Mark axes and class boundaries accurately; cumulative frequency cannot decrease as the measured value increases.
Read a cumulative position across to the curve
For n observations, locate a requested percentile's cumulative position on the vertical axis, read horizontally to the curve, then down to the measurement axis. The median is the 50th percentile; quartiles are the 25th and 75th percentiles.
Estimate IQR = Q3 − Q1. Because the graph is based on grouped data, readings are estimates; individual values within each interval are not known.
Pause and explain
How is the interquartile range calculated?
Worked example
A cumulative frequency graph represents 80 journey times. At cumulative frequency 40, the curve gives 24 minutes; at 20 it gives 18 minutes and at 60 it gives 31 minutes. Estimate the median, lower and upper quartiles, and IQR.
Show the worked solution
- The median is at n/2 = 40, so it is about 24 minutes.
- The lower and upper quartile positions are n/4 = 20 and 3n/4 = 60, giving about 18 and 31 minutes.
- IQR ≈ 31 − 18 = 13 minutes.
Answer Median ≈ 24 min; Q1 ≈ 18 min; Q3 ≈ 31 min; IQR ≈ 13 min.
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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.
A cumulative frequency curve describes 100 test times. Estimate the 25th percentile, median and 75th percentile at cumulative frequencies 25, 50 and 75.
Give me a hint
Use the corresponding cumulative counts, then read from the curve to the time axis.
Compare my reasoning
- At CF 25, the curve gives about 12 min.
- At CF 50, it gives about 16 min; at CF 75, about 21 min.
- The estimates are Q1 ≈ 12 min, median ≈ 16 min and Q3 ≈ 21 min.
Q1 ≈ 12 min; median ≈ 16 min; Q3 ≈ 21 min.
Look for these in your work
- Correct percentile positions used
- Readings taken from curve
- Estimates labelled
For the estimates Q1 = 12, median = 16 and Q3 = 21 minutes, find the IQR and describe its meaning.
Give me a hint
Subtract the lower quartile from the upper quartile.
Compare my reasoning
- IQR = 21 − 12.
- IQR = 9 minutes.
- The middle half of observed times spans about 9 minutes.
IQR ≈ 9 minutes; the middle half spans about 9 minutes.
Look for these in your work
- Quartile difference calculated
- Units stated
- Spread interpreted
A cumulative frequency graph is based on 200 grouped measurements. The curve reads 42 at 18 units and 166 at 30 units. Estimate how many measurements are from 18 to 30 units, and state why the result may be approximate.
Give me a hint
Subtract cumulative counts at the two boundaries.
Compare my reasoning
- Up to 30 units: CF ≈ 166.
- Up to 18 units: CF ≈ 42.
- Between 18 and 30 units: 166 − 42 = about 124; curve readings and grouping make it an estimate.
About 124 measurements; grouped curve readings are estimates.
Look for these in your work
- Cumulative values differenced
- Interval count interpreted
- Approximation acknowledged
Common mistakes
- Using frequency rather than cumulative frequency for a plotted point
- Reading a percentile from the horizontal axis first
- Forgetting the cumulative total equals n
- Presenting a graph estimate as an exact individual-data value
What can you explain now?
For 120 values, at what cumulative frequency do you read the median?
Compare with the explanation
60
The median is the 50th percentile, n/2.
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Make it stick
How do cumulative counts let you estimate quartiles and spread, and why are the readings approximate?
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.