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Cumulative Frequency Graphs and Box Plots Explained

Find the median, quartiles and interquartile range, draw box plots (box and whisker plots) and compare data sets, with worked examples.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Intermediate
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Curriculum fit: GCSE Maths Statistics (mainly Higher tier). US: High School Statistics (box plots, interquartile range).

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What Is It?

Quartiles divide ordered data into four equal parts. The lower quartile (LQ) has 25% of values below it, the median has 50% and the upper quartile (UQ) has 75%. The interquartile range (IQR) = UQ − LQ measures the spread of the middle half of the data.

A cumulative frequency graph adds up frequencies as you go along. You can read the median and quartiles from it. A box plot shows the minimum, LQ, median, UQ and maximum.

min 2Q1 5median 8Q3 12max 18IQR = 12 − 5 = 705101520range = 18 − 2 = 16
A box plot shows the five-number summary. The box holds the middle half of the data, and the interquartile range (IQR) is the width of the box.

Key Formulas and Ideas

  • Five-number summary: minimum, LQ, median, UQ, maximum.
  • IQR = UQ − LQ. Range = maximum − minimum.
  • On a cumulative frequency graph with n values: median at n/2, LQ at n/4, UQ at 3n/4.

Worked Examples

Example 1

Quartiles From a List

Question

Find the median, LQ, UQ and IQR of 3, 5, 7, 8, 9, 11, 13, 14, 16, 19, 20.

  1. 11 values, so the median is the 6th value: 11.
  2. Lower half (excluding the median): 3, 5, 7, 8, 9, so LQ = 7.
  3. Upper half: 13, 14, 16, 19, 20, so UQ = 16.
  4. IQR = 16 − 7 = 9.

✓ Answer Median 11, LQ 7, UQ 16, IQR 9

Example 2

Cumulative Frequency

Question

A cumulative frequency graph is based on 80 values. Where do you read the median?

  1. Median is at n/2 = 40.
  2. Go across from 40 on the vertical axis to the curve, then down.

✓ Answer At cumulative frequency 40

Example 3

Comparing

Question

Class A has median 62 and IQR 10. Class B has median 58 and IQR 22. Compare them.

  1. Class A has the higher median, so typically scores higher.
  2. Class A has the smaller IQR, so its scores are more consistent.

✓ Answer Class A: higher and more consistent

Common Mistakes to Avoid

  • Plotting cumulative frequency at the class start instead of the upper class boundary.
  • Joining points with straight lines in a box plot instead of drawing a box.
  • Including the median when splitting the data into halves for an odd number of values.
  • Comparing only averages and ignoring spread.

Practice Questions

0 of 5 answered
Q1For data 2, 4, 6, 8, 10, what is the median?
Q2LQ = 12 and UQ = 20. What is the IQR?
Q3With 100 values, where is the median read on a cumulative frequency graph?
Q4A box plot shows five values. Which is NOT one of them?
Q5A smaller IQR means the data is…

Frequently Asked Questions

What is an outlier in a box plot?

A value far from the rest. A common rule marks a value more than 1.5 × IQR beyond a quartile. Check your exam board's rule.

Why use the IQR instead of the range?

The IQR ignores extreme values, so it is less affected by outliers.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.