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Cumulative Frequency Diagrams

FoundationHigherAQAEdexcel

Master Cumulative Frequency Diagrams for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Cumulative frequency diagrams plot running totals and are used to estimate the median and quartiles.

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

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A cumulative frequency diagram shows running totals, allowing the median and quartiles to be estimated from grouped data.

The cumulative frequency is found by adding each frequency to the running total. Each point is plotted at the upper class boundary, not the midpoint, because the running total is only complete at the end of the interval. The points are joined with a smooth curve.

To estimate the median, read across from half the total frequency. The lower quartile is at a quarter and the upper quartile at three quarters. Plotting at the midpoint instead of the upper boundary is the most common error and shifts every reading.

Revision notes

Building the table

Add each frequency to the running total to get the cumulative frequency.

The final cumulative frequency equals the total frequency, which provides a check. Each cumulative value tells you how many items are less than the upper boundary of that class.

Plotting the curve

Plot each cumulative frequency against the UPPER class boundary, not the midpoint.

This is because the running total is only complete once the whole interval has been included. Join the points with a smooth curve.

Reading off values

Median: read across from \(\frac{n}{2}\) on the cumulative frequency axis.

Lower quartile: read across from \(\frac{n}{4}\). Upper quartile: read across from \(\frac{3n}{4}\). Interquartile range is the upper quartile minus the lower quartile.

Key points

  • Cumulative frequency is a running total.
  • Plot against the upper class boundary.
  • The final value equals the total frequency.
  • Join the points with a smooth curve.
  • The median is read from half the total.
  • Quartiles are read from a quarter and three quarters.

Worked examples

Example 1

Frequencies are 5, 8 and 11. Work out the cumulative frequencies. [3 marks]

Working

First \(= 5\)the first cumulative frequency is the first frequency
\(5 + 8 = 13\)add the second frequency to the running total
\(13 + 11 = 24\)add the third frequency

Example 2

A cumulative frequency diagram has a total frequency of 80. State the cumulative frequency used to estimate the median. [2 marks]

Working

\(\frac{80}{2}\)halve the total frequency
\(= 40\)read across from 40 on the cumulative frequency axis

Example 3

Explain why points are plotted at the upper class boundary. [2 marks]

Working

The cumulative frequency counts all values up to the end of that classstate what it represents
so the running total is only complete at the upper boundary, not at the midpointexplain why that position is used

Common mistakes

  • Plotting at the midpoint.

    Cumulative frequency points go at the upper class boundary.

  • Reading the median from the horizontal axis.

    Read across from half the total on the vertical axis, then down.

  • Not checking the final cumulative frequency.

    It must equal the total frequency.

  • Joining points with straight lines.

    A smooth curve is expected.

Exam tips

  • Plot at the upper class boundary every time.
  • Check the final cumulative frequency equals the total.
  • Read across from n/2, n/4 and 3n/4.
  • Draw a smooth curve, not straight segments.

Key terms

Cumulative frequency
A running total of frequencies.
Upper class boundary
The top value of a class, where points are plotted.
Median
The middle value, read from half the total.
Quartile
A value dividing the data into quarters.

Related topics

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