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Averages from frequency tables

Use frequency tables to find the mean, median, mode and range without writing out the whole list unless it helps.

Probability and StatisticsFoundation and HigherGrades 3 to 5Focused skill

Curriculum path: GCSE Maths > Edexcel > Probability and Statistics > Averages from frequency tables

Pearson Edexcel GCSE Maths Statistics S2 and S4: interpret frequency tables and calculate averages and spread.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A frequency tells you how many times a value appears. In a frequency table, each row stands for repeated values.

For the mean, multiply each value by its frequency. Add these products, then divide by the total frequency.

For the median, use the total frequency to find the middle position. Then use cumulative frequency to locate that position in the table.

For the mode, choose the value with the highest frequency.

For the range, use the largest value with a frequency and subtract the smallest value with a frequency.

If students are unsure, expanding the table into a list can help for small frequencies, but it is too slow for larger exam questions.

Key ruleMean from a frequency table = total of value x frequency / total frequency.

Diagram guide

Score1234Frequency2541Score x f210124mean = total score x f / total f
Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.

Worked examples

Mean from a frequency table

Use the table to find the mean score: scores 1, 2, 3, 4 have frequencies 2, 5, 4, 1.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Score frequency tableThe frequency row tells you how many students got each score.
  1. Find score x frequency: 1 x 2 = 2, 2 x 5 = 10, 3 x 4 = 12, 4 x 1 = 4.
  2. Total score = 2 + 10 + 12 + 4 = 28.
  3. Total frequency = 2 + 5 + 4 + 1 = 12.
  4. Mean = 28 / 12 = 2.333...

Answer: 2.33 to 3 significant figures

Median from a frequency table

Find the median score from the same table.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  1. There are 12 scores, so the median is halfway between the 6th and 7th values.
  2. Cumulative frequency after score 1 is 2.
  3. Cumulative frequency after score 2 is 7, so the 6th and 7th values are both 2.

Answer: 2

Mode and range

Find the mode and range from the same table.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  1. The highest frequency is 5, which belongs to score 2.
  2. The smallest score is 1 and the largest score is 4.
  3. Range = 4 - 1 = 3.

Answer: Mode = 2, range = 3

Common mistakes

  • Adding the frequencies and calling that the mean.
  • Forgetting the value x frequency column.
  • Using the highest frequency as the mode instead of the value with the highest frequency.
  • Finding the median position but not using cumulative frequency to locate it.
  • Including a value with frequency 0 when finding the range.

Quick exercise

Try these before moving to the exam-style questions.

  1. Values 1, 2, 3 have frequencies 4, 3, 1. Find the mean.
    Score1234Frequency2541Score x f210124mean = total score x f / total f
    Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  2. Values 2, 4, 6 have frequencies 1, 5, 2. Find the mode.
    Score1234Frequency2541Score x f210124mean = total score x f / total f
    Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  3. Values 0, 1, 2, 3 have frequencies 2, 4, 3, 1. Find the median.
    Score1234Frequency2541Score x f210124mean = total score x f / total f
    Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  4. Values 5, 6, 7, 8 have frequencies 0, 3, 2, 1. Find the range.
    Score1234Frequency2541Score x f210124mean = total score x f / total f
    Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
  5. Why is value x frequency needed for the mean?
    Score1234Frequency2541Score x f210124mean = total score x f / total f
    Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationCalculator4 marks

A group of students scored 0, 1, 2 or 3 in a quiz. The frequencies were 1, 4, 3 and 2. Find the mean score.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Question tableUse value x frequency, then divide by the total frequency.
frequency tablemeanfoundation statistics
Standard exam styleFoundation and HigherEither4 marks

The table shows the number of pets owned by 15 students. Number of pets: 0, 1, 2, 3. Frequency: 3, 6, 4, 2. Find the median and the mode.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
medianmodecumulative frequency
ChallengeFoundation and HigherCalculator5 marks

A frequency table has values 4, 5, 6 and 7. The frequencies are 2, x, 5 and 1. The total frequency is 12. Work out the mean.

Score1234Frequency2541Score x f210124mean = total score x f / total f
Discrete frequency tableMultiply each score by its frequency for the mean, and use cumulative positions for the median.
frequency tablemissing frequencymean