Skip to content
VirtusAcademy

Averages from Frequency Tables

FoundationHigherAQAEdexcel

Master Averages from Frequency Tables for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. The mean, median and mode can be found from a frequency table using the frequencies.

Free downloads

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

When data is presented in a frequency table, the averages are found using the frequencies rather than by listing every value out.

For the mean, multiply each value by its frequency, add these products, then divide by the total frequency. This gives \(\frac{\sum fx}{\sum f}\).

For the median, find the position \(\frac{n+1}{2}\) where \(n\) is the total frequency, then use a running total of frequencies to locate which value sits in that position. The mode is the value with the highest frequency — read from the frequency column, not the value column.

Revision notes

The mean from a frequency table

Multiply each value \(x\) by its frequency \(f\) to get \(fx\).

Add the \(fx\) column to get \(\sum fx\). Add the frequency column to get \(\sum f\). Then mean \(= \frac{\sum fx}{\sum f}\).

The median from a frequency table

Find the position of the median using \(\frac{n+1}{2}\), where \(n\) is the total frequency.

Work through a running total of the frequencies until you reach or pass that position. The value at that point is the median.

The mode from a frequency table

The mode is the value with the highest frequency.

Read across from the largest number in the frequency column to the corresponding value. Giving the frequency itself as the answer is a frequent error.

Key points

  • Multiply each value by its frequency.
  • Mean = sum of fx ÷ sum of f.
  • The median position is (n+1) ÷ 2.
  • Use a running total to locate the median.
  • The mode is the value with the highest frequency.
  • Do not give the frequency as the mode.

Worked examples

Example 1

Values 1, 2 and 3 have frequencies 4, 6 and 10. Work out the mean. [3 marks]

Working

\(fx\) values: \(1 \times 4 = 4\), \(2 \times 6 = 12\), \(3 \times 10 = 30\)multiply each value by its frequency
\(\sum fx = 4 + 12 + 30 = 46\) and \(\sum f = 4 + 6 + 10 = 20\)find both totals
\(\frac{46}{20} = 2.3\)divide to find the mean

Example 2

A frequency table has a total frequency of 39. Work out the position of the median. [2 marks]

Working

\(\frac{39 + 1}{2}\)use the median position formula
\(= 20\), so the median is the 20th valuework out the position

Example 3

Values 5, 6 and 7 have frequencies 3, 11 and 8. State the mode. [1 mark]

Working

6the value with the highest frequency of 11

Common mistakes

  • Giving the frequency as the mode.

    The mode is the VALUE that occurs most often, not how often it occurs.

  • Dividing by the number of rows instead of the total frequency.

    Divide by the sum of the frequencies.

  • Forgetting to multiply value by frequency.

    The fx column is essential for the mean.

  • Using n ÷ 2 for the median position.

    The position is (n+1) ÷ 2.

Exam tips

  • Add an fx column to the table before calculating.
  • Divide by the total frequency, not the number of rows.
  • Use (n+1) ÷ 2 for the median position.
  • Read the mode from the value column, not the frequency column.

Key terms

Frequency table
A table showing how often each value occurs.
\(\sum fx\)
The total of value multiplied by frequency.
\(\sum f\)
The total frequency.
Running total
A cumulative sum used to locate the median.

Related topics

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