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GCSE averages practice and worked examples

Compare mean, median and mode, and use the range to describe spread. Extend the method to data given in frequency tables.

Foundation and Higher · Useful for students building fluency and applying the method in GCSE questions.

Core pathway. Core GCSE methods are useful across Foundation and Higher. Start with the core examples and questions.

Higher extension. More demanding selections are marked Higher extension: practice question 4. Use these after the core work, following your teacher’s guidance.

4 worked examples and 8 original practice questions · Allow 35–50 minutes · Read online or print. Higher extensions are labelled. No account or email needed.

Learn the methodTry the questions

Printing includes the method, examples and space for working, followed by a separate answer section.

A useful approach

  1. Sort data before finding the median; check whether there are one or two middle values.
  2. For a mean, divide the total by the number of values. In a frequency table, multiply each value by its frequency first.
  3. Name the average requested. Find the range by subtracting the smallest value from the largest.

A mistake to watch for

The number of rows in a frequency table is not the number of observations. A grouped-table mean is an estimate when class midpoints are used.

Learn each method, then practise it

Read each line of working and explain why it follows from the previous line.

Mean from a list

Add all values and divide by how many there are. Keep the units, converting at the end when requested.

Worked example 1

Averages · Core GCSE skill · No calculator · 2 marks

Here are four numbers.

6  11  9  14

Work out the mean of these numbers.

  1. The mean is the amount each value would have if the total were shared equally. Add all the values, then divide by how many values there are.
  2. Total: 6 + 11 + 9 + 14 = 40.
  3. Mean: 40 divided by 4 = 10.
Answer: 10
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Now practise this method: Question 1 · Question 2

Mode and range

The mode is the most frequent value, not the frequency itself. The range is the largest value minus the smallest.

Worked example 2

Averages · Core GCSE skill · No calculator · 1 marks

Here are seven numbers.

3  7  7  4  9  7  2

Write down the mode of these numbers.

  1. The mode is the value that appears most often.
  2. 7 appears three times, more than any other number, so the mode is 7.
Answer: 7
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Now practise this method: Question 3

Median and spread

Order the values first. Find the middle position for the median, and largest minus smallest for the range. Compare both typical value and spread.

Worked example 3

Averages · Core GCSE skill · Calculator allowed · 4 marks

Nine students in Year 7 and nine students in Year 11 ran a race. Here are their times, in seconds.

Year 7:  12  15  15  18  20  21  24  26  28

Year 11:  8  10  13  14  16  19  22  25  27

Compare the times of the Year 7 students with the times of the Year 11 students.

  1. Year 7 median: the 5th value = 20 seconds. Year 11 median: the 5th value = 16 seconds.
  2. Year 7 range: 28 - 12 = 16 seconds. Year 11 range: 27 - 8 = 19 seconds.
  3. The Year 11 median is lower, so on average the Year 11 students ran faster.
  4. The Year 7 range is smaller, so the Year 7 times were more consistent.
Answer: Year 11 median (16 s) is lower than Year 7 median (20 s), so Year 11 were faster on average; Year 7 range (16 s) is smaller than Year 11 range (19 s), so Year 7 times were more consistent
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Now practise this method: Question 4

Frequency tables

Frequencies count observations. For a mean multiply each value by its frequency, add those products and divide by the total frequency. For the mode, choose the value with the highest frequency. For the median, use cumulative counts to locate the middle observation(s). A missing frequency can be found by forming an equation from the stated mean.

Worked example 4

Averages from Frequency Tables · Core GCSE skill · No calculator · 3 marks

The table shows the number of children in each of 20 families.

Number of childrenFrequency
03
18
26
33

Work out the mean number of children per family.

  1. Total children = (0 × 3) + (1 × 8) + (2 × 6) + (3 × 3) = 0 + 8 + 12 + 9 = 29.
  2. Total families = 20.
  3. Mean = 29 ÷ 20 = 1.45.
Answer: 1.45
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Now practise this method: Question 5 · Question 6 · Question 7 · Question 8

Your practice questions

Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.

Question 5

Averages from Frequency Tables · Core GCSE skill · Calculator allowed · 2 marks

The table shows the number of goals scored by a hockey team in each of its matches last season.

Number of goalsFrequency
07
111
28
34
  1. Write down the number of matches the team played.
  2. Ria says, "The median number of goals is 1.5, because 1.5 is halfway between 0 and 3." Explain why Ria is wrong.

Check answer and working for question 5

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Question 6

Averages from Frequency Tables · Core GCSE skill · Calculator allowed · 3 marks

The table shows the shoe sizes of the students in a year group.

Shoe sizeFrequency
45
59
612
78
86
  1. Write down the modal shoe size.
  2. Work out the number of students in the year group.
  3. Work out how many students have a shoe size of 7 or larger.

Check answer and working for question 6

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Question 8

Averages from Frequency Tables · Core GCSE skill · Calculator allowed · 4 marks

The table shows the number of items bought by each of 40 customers at a corner shop.

Number of itemsFrequency
112
215
39
44
  1. Work out the mean number of items bought per customer.
  2. Kai says, "The mode is 15." Explain the mistake Kai has made.

Check answer and working for question 8

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Worked answers: Averages

Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.

Answer 1

Show answer and working for question 1
  1. Total: 210 + 185 + 240 + 195 + 230 = 1060 seconds.
  2. Mean: 1060 divided by 5 = 212 seconds.
  3. 212 seconds = 180 seconds + 32 seconds = 3 minutes 32 seconds.
Answer: 3 minutes 32 seconds
Back to question 1

Answer 2

Show answer and working for question 2
  1. Total of all six scores: 6 × 15 = 90.
  2. Total of the first five scores: 12 + 18 + 9 + 21 + 14 = 74.
  3. Sixth score: 90 - 74 = 16.
Answer: 16
Back to question 2

Answer 3

Show answer and working for question 3
  1. a) Size 5 appears three times, more than any other size, so the mode is 5.
  2. b) Range: 9 - 5 = 4.
Answer: (a) 5   (b) 4
Back to question 3

Answer 4

Show answer and working for question 4
  1. Mean 6 means the total is 5 × 6 = 30.
  2. Write the numbers in order: a, b, c, d, e. The median gives c = 5.
  3. The mode is 4, so 4 must appear at least twice. Both 4s must be below the median, so a = 4 and b = 4.
  4. Range 7 gives e - a = 7, so e = 4 + 7 = 11.
  5. Total: 4 + 4 + 5 + d + 11 = 30, so d = 6.
  6. Check: numbers 4, 4, 5, 6, 11 have mean 6, mode 4, median 5, range 7.
Answer: 4, 4, 5, 6, 11
Back to question 4

Answer 5

Show answer and working for question 5
  1. Total matches = 7 + 11 + 8 + 4 = 30.
  2. The cumulative frequencies are 7, 18, 26, 30.
  3. The 15th and 16th values both lie in the group of matches with 1 goal, so the median is 1, not 1.5.
Answer: (a) 30   (b) The median must come from the ordered data values, not halfway between the smallest and largest; the 15th and 16th values are both 1, so the median is 1
Back to question 5

Answer 6

Show answer and working for question 6
  1. The mode is the shoe size with the highest frequency: size 6 (frequency 12).
  2. Total students = 5 + 9 + 12 + 8 + 6 = 40.
  3. Size 7 or larger = 8 + 6 = 14.
Answer: (a) 6   (b) 40   (c) 14
Back to question 6

Answer 7

Show answer and working for question 7
  1. 6 + 10 + 7 + 4 = 27.
  2. x = 35 - 27 = 8.
  3. The highest frequency is 10, for 1 book, so the mode is 1.
Answer: (a) 8   (b) 1
Back to question 7

Answer 8

Show answer and working for question 8
  1. Total items = (1 × 12) + (2 × 15) + (3 × 9) + (4 × 4) = 12 + 30 + 27 + 16 = 85.
  2. Mean = 85 ÷ 40 = 2.125.
  3. The mode is the number of items with the highest frequency, so the mode is 2, not 15.
Answer: (a) 2.125   (b) Kai has given the largest frequency instead of the data value; the mode is 2
Back to question 8

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