Cumulative frequency curves and percentiles
Accumulate frequencies at upper boundaries, then estimate medians, quartiles and percentiles from a curve.
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
- Read grouped class boundaries.
- Interpret quartiles and percentages.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Where should a cumulative-frequency class total be plotted?
What you will learn
- Construct a cumulative frequency table and curve.
- Read percentile values using the total frequency.
The running total is plotted at upper boundaries
For classes 0 ≤ x < 10, 10 ≤ x < 20, 20 ≤ x < 30 with frequencies 4,8,8, cumulative frequencies are four, twelve and twenty. Plot (0,0), (10,4), (20,12), (30,20) on value and cumulative-frequency axes, then draw a smooth increasing curve through the points.
Use upper class boundaries, not midpoints. The graph cannot decrease because additional classes add non-negative counts. The last cumulative frequency equals the sample size twenty. The initial lower boundary with zero provides the starting point, and all plotted points should remain clear.
Read the value corresponding to a proportion of the total
For twenty observations, read the median at cumulative frequency ten, Q1 at five, Q3 at fifteen, and the 90th percentile at eighteen. Move horizontally from the cumulative-frequency position to the curve, then vertically to the value axis. Quartiles are measured values, not the frequency positions themselves.
Graph readings are estimates because grouping hides individual observations. A straight within-class interpolation gives median 17.5, Q1 11.25 and Q3 23.75 for this example; a hand-drawn smooth curve may give slightly different estimates. Label the method and appropriate accuracy rather than claiming exact original values.
Pause and explain
There are 20 observations. At what cumulative frequency do you read Q3?
Worked example
For grouped frequencies 4,8,8 in 0–10,10–20,20–30, estimate the median by linear within-class interpolation.
Show the worked solution
- Total N = 20, so use cumulative position N/2 = 10.
- The median lies in 10–20, where the cumulative count rises from four to twelve.
- Move (10−4)/8 of the class width: 10 + (6/8)×10 = 17.5.
Answer Estimated median 17.5 using within-class interpolation.
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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.
Class frequencies are 3,7,5,5. Find cumulative frequencies.
Give me a hint
Keep adding each new class to the running total.
Compare my reasoning
- First total is three; second is ten.
- Third total is fifteen; fourth is twenty.
- The final count equals all twenty observations.
3,10,15,20.
Look for these in your work
- Used running totals.
- Checked the final sample size.
A cumulative curve has total 80. State frequency positions for median, Q1, Q3 and 90th percentile.
Give me a hint
Apply one half, one quarter, three quarters and ninety percent to N.
Compare my reasoning
- Median position is forty and Q1 is twenty.
- Q3 position is sixty.
- The 90th-percentile position is seventy-two.
40,20,60,72 respectively.
Look for these in your work
- Used the total frequency.
- Separated positions from data values.
Using linear interpolation for the 4,8,8 example, estimate Q1, Q3 and IQR.
Give me a hint
Positions five and fifteen lie in different classes.
Compare my reasoning
- Q1 = 10 + (5−4)/8×10 = 11.25.
- Q3 = 20 + (15−12)/8×10 = 23.75.
- Estimated IQR = 23.75−11.25 = 12.5.
Q1 ≈ 11.25; Q3 ≈ 23.75; IQR ≈ 12.5.
Look for these in your work
- Interpolated within the correct classes.
- Labelled grouped readings as estimates.
Common mistakes
- Plotting totals at midpoints.
- Giving a frequency position as the measured median.
- Calling grouped estimates exact.
What can you explain now?
Can a valid cumulative-frequency curve decrease?
Compare with the explanation
No.
Each step adds a non-negative count, so cumulative frequency never falls.
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Make it stick
Tomorrow, build a running-total table and explain both axes before reading one percentile.
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.