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Cumulative Frequency

HigherHigher tier onlyAQAEdexcelOCR

Cumulative Frequency is a key statistics topic at GCSE Maths. This Higher worksheet gives you exam-style questions on cumulative frequency graphs, medians and quartiles, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Plot the running total at the upper boundary of each class.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A cumulative frequency diagram shows running totals, letting you estimate the median and quartiles from grouped data.

Each point is plotted at the upper boundary of its class against the running total up to that point. Plotting at the midpoint is wrong here — that belongs to frequency polygons — because the running total is only complete at the end of the class.

The points are joined with a smooth curve. To find the median, read across from half the total frequency; for the quartiles, read across from one quarter and three quarters.

Revision notes

Building the table

Add a cumulative frequency column, keeping a running total down the classes.

For frequencies \(5, 8, 12, 5\), the cumulative values are \(5, 13, 25, 30\). The final value equals the total frequency.

Plotting the points

Plot each cumulative frequency against the upper boundary of its class.

For the class \(10 \le x < 20\) with cumulative frequency \(13\), plot \((20, 13)\). Join the points with a smooth curve.

Reading the median and quartiles

The median is at half the total, the lower quartile at a quarter and the upper quartile at three quarters.

With a total of \(40\): read across from \(20\) for the median, \(10\) for \(Q_1\) and \(30\) for \(Q_3\). Leave your reading lines visible.

Key points

  • Cumulative frequency is a running total.
  • Plot against the upper class boundary.
  • Join the points with a smooth curve.
  • The median is at half the total frequency.
  • \(Q_1\) is at a quarter, \(Q_3\) at three quarters.
  • Show your reading lines on the diagram.

Worked examples

Example 1

Frequencies are \(6, 9, 10, 5\). Find the cumulative frequencies.

Working

\[6, 6+9 = 15\]keep a running total
\[15+10 = 25, 25+5 = 30\]continue down the classes
\[6, 15, 25, 30\]state the cumulative frequencies

Example 2

A total of \(60\) is shown. Find the cumulative frequency to read across from for the median.

Working

\[60 \div 2\]the median is at half the total
\[= 30\]read across from 30

Example 3

For a total of \(80\), find the values for \(Q_1\) and \(Q_3\).

Working

\[80 \div 4 = 20\]the lower quartile is at a quarter
\[3 \times 20 = 60\]the upper quartile is at three quarters

Common mistakes

  • Plotting at the class midpoint.

    Cumulative frequency is plotted at the upper class boundary, not the midpoint.

  • Joining the points with straight lines.

    A smooth curve is expected for a cumulative frequency diagram.

  • Using \(\frac{n+1}{2}\) for the median position.

    On a cumulative frequency graph, read across from half the total, not from (n+1)/2.

  • Not showing reading lines.

    Method marks are awarded for the lines drawn across and down.

Exam tips

  • Add a cumulative frequency column before plotting.
  • Plot at the upper boundary of each class.
  • Draw a smooth curve through the points.
  • Leave your reading lines on the diagram.

Key terms

Cumulative frequency
A running total of frequencies.
Upper boundary
The top value of a class.
Median
The middle value, at half the total frequency.
Quartile
A value dividing the data into quarters.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.