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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.

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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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