Rukasha Maths

GCSE maths Beta

Cumulative frequency and box plots · Higher notes

Higher · Statistics · AQA references S3, S4

Cumulative frequency counts how many observations are below each upper class boundary. Add frequencies progressively and plot these running totals against the upper boundaries. A cumulative frequency graph estimates quartiles by locating one quarter, one half and three quarters of the total frequency on the vertical axis. A box plot summarises the minimum, quartiles, median and maximum. Compare typical values with medians and compare spread with interquartile ranges. Graph estimates depend on grouped data, so they are not exact original measurements.

Learning objectives

  • Estimate a median and quartiles from cumulative frequency.
  • Compare the centre and spread of distributions using box plots.

Key ideas

Cumulative frequency is a running total; plot it against upper class boundaries.

For grouped graphs, locate quartiles using these cumulative frequency positions.

Q1:N4,median:N2,Q3:3N4Q_1:\frac N4,\qquad\text{median}:\frac N2,\qquad Q_3:\frac{3N}{4}

The interquartile range measures the spread of the middle half.

IQR=Q3−Q1\mathrm{IQR}=Q_3-Q_1

A box plot shows minimum, lower quartile, median, upper quartile and maximum. Compare centre and spread separately.

Worked examples

Example 1

For 80 journeys, readings at cumulative frequencies 20, 40 and 60 are 18, 26 and 34 minutes. Estimate the median and interquartile range.

804=20,802=40,3(80)4=60\frac{80}{4}=20,\qquad\frac{80}{2}=40,\qquad\frac{3(80)}4=60
Q1≈18,median≈26,Q3≈34 minutesQ_1\approx18,\qquad\text{median}\approx26,\qquad Q_3\approx34\ \text{minutes}
IQR≈34−18\mathrm{IQR}\approx34-18

Answer

median≈26 minutes,IQR≈16 minutes\text{median}\approx26\ \text{minutes},\qquad\mathrm{IQR}\approx16\ \text{minutes}

Example 2

The lower quartile is sixteen and the upper quartile twenty-nine. Find the interquartile range.

IQR=Q3−Q1=29−16\mathrm{IQR}=Q_3-Q_1=29-16

Answer

1313

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Read quartiles or a median and draw a box plot or cumulative frequency graph.

AO2 · Reasoning

Compare distributions using median and interquartile range in context.

AO3 · Problem solving

Estimate a count above a threshold and explain the limitation of grouped data.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Read the median at half the total cumulative frequency. Graph readings are estimates; interquartile range is not the full range.

Try it yourself

Attempt each question before opening its answer.

Check 1: A cumulative frequency graph has 60 observations. At which cumulative frequency do you read the median?

Answer: 30

Half the observations lie below the median: N/2 = 30.

Check 2: Q1 = 16 and Q3 = 29. Find the IQR.

Answer: 13

Subtract the lower quartile from the upper quartile.

Explain it in your own words

  • Explain what the word cumulative means in a frequency graph.
  • Write the three quartile positions for N = 120 observations.
  • Compare two distributions using a median statement and an IQR statement.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Statistics specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.