EST. 2024 · LONDON·MMXXVI SPECIFICATION
AQA·Edexcel·OCR|Foundation + Higher
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Statistics & Probability · Higher

Cumulative frequency diagrams

Cumulative frequency diagrams show the running total of frequencies. They are used to estimate the median, quartiles and interquartile range (IQR) of grouped data.

Revising cumulative frequency diagrams the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Cumulative frequency diagrams GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Cumulative frequency = running total of frequencies.
  • 2Plot cumulative frequency against the upper class boundary.
  • 3Join points with a smooth S-shaped curve.
  • 4Median: read across from n/2 on the CF axis.
  • 5Lower quartile (Q1): read from n/4; upper quartile (Q3): read from 3n/4.
  • 6IQR = Q3 − Q1.
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Formulas

Median position
n/2 th value
IQR
IQR = Q3 − Q1
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Worked examples

Example 1

From a cumulative frequency graph with n = 80, read off the median, Q1 and Q3.

Working

  1. Median: read across from CF = 40 → read corresponding x-value
  2. Q1: read across from CF = 20 → read corresponding x-value
  3. Q3: read across from CF = 60 → read corresponding x-value
  4. IQR = Q3 − Q1
AnswerRead from graph: Median at CF = 40, Q1 at CF = 20, Q3 at CF = 60
Example 2

Masses (kg): 0-20(f=5), 20-40(f=15), 40-60(f=25), 60-80(f=10), 80-100(f=5). Draw the cumulative frequency table.

Working

  1. CF: 0→0, 20→5, 40→20, 60→45, 80→55, 100→60
  2. Plot points at upper boundaries: (20,5), (40,20), (60,45), (80,55), (100,60)
  3. Join with smooth curve
AnswerCF points: (20,5), (40,20), (60,45), (80,55), (100,60)
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Common mistakes

✗Plotting against the midpoint instead of the upper class boundary.
✗Reading the median at n instead of n/2.
✗Drawing straight lines between points rather than a smooth curve.
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Exam tips

✓Always plot against the upper boundary, not the midpoint.
✓Draw a horizontal line from the CF axis to the curve, then a vertical line down to read the x-value.

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