Averages and range | AQA GCSE Maths Notes

Visual model

Mean, median, mode and range tell different stories

medianrange = biggest - smallest

Gold-standard guide

20 mins

What you will learn

Calculate and compare mean, median, mode and range.

Use a clear step-by-step method for averages and range.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier statistics skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Mean = sum / count

Step 1

Order the data and find the range

Ordered: 3, 4, 5, 7, 8, 8, 11

Step 2

Find the median

7 values — middle is the 4th: median = 7

Step 3

Find the mode

8 appears twice — all others once

Watch out

Students find the median without sorting the data first, $pic$ king a middle-looking value from the unsorted list

Mean

$mean = tota\frac{l}{number} of values.$

Range

$range = highest - lowest.$

Worked example

Find the mean, median, mode and range of: 4, 8, 8, 5, 11, 7, 3.

Order the data and find the range: Ordered: 3, 4, 5, 7, 8, 8, 11. Range = 11 − 3 = 8.

Find the median: 7 values — middle is the 4th: median = 7.

Find the mode: 8 appears twice — all others once. Mode = 8.

Find the mean: Sum = 3 + 4 + 5 + 7 + 8 + 8 + 11 = 46. Mean = $4\frac{6}{7}$ ≈ 6.57.

Final answer

Mean ≈ 6.57; Median = 7; Mode = 8; Range = 8

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Find the mean, median, mode and range of: 4, 8, 8, 5, 11, 7, 3.

4 marks4 minsaverages-and-range-worked

Show solution

Worked solution

1.Order the data and find the range: Ordered: 3, 4, 5, 7, 8, 8, 11. Range = 11 − 3 = 8.
2.Find the median: 7 values — middle is the 4th: median = 7.
3.Find the mode: 8 appears twice — all others once. Mode = 8.
4.Find the mean: Sum = 3 + 4 + 5 + 7 + 8 + 8 + 11 = 46. Mean = $4\frac{6}{7}$ ≈ 6.57.

Final answer

Mean ≈ 6.57; Median = 7; Mode = 8; Range = 8

Mark points

M1: order the data and find the range
M1: find the median
M1: find the mode
M1: find the mean
A1: Mean ≈ 6.57; Median = 7; Mode = 8; Range = 8

Watch out

Students find the median without sorting the data first, $pic$ king a middle-looking value from the unsorted list.Always sort. With an even number of values, the median is the mean of the two middle values.

Diagnosticrecall

Find the median of: 2, 5, 8, 12, 15, 19.

1 mark2 minsaverages-and-range-q1

Show solution

Worked solution

1.Spot the skill: Mean = sum / count.
2.Use the order the data and find the range stage first, then find the median.
3.Keep the final answer visible: 10 (mean of 8 and 12).

Final answer

10 (mean of 8 and 12)

Mark points

M1: use the correct mean = sum / count. median = middle value when ordered. mode = most frequent value.range = largest − smallest. the mean is sensitive to outliers; the median is more robust.use the appropriate average for the context.
A1: 10 (mean of 8 and 12)

Watch out

Easyprocedure

Find the mean of: 6, 6, 7, 9, 12.

2 marks3 minsaverages-and-range-q2

Show solution

Worked solution

1.Spot the skill: Mean = sum / count.
2.Use the find the median stage first, then find the mode.
3.Keep the final answer visible: 8.

Final answer

Mark points

M1: use the correct mean = sum / count. median = middle value when ordered. mode = most frequent value.range = largest − smallest. the mean is sensitive to outliers; the median is more robust.use the appropriate average for the context.
A1: 8

Watch out

Mediumreasoning

The mean of 5 numbers is 12. Four of them are 8, 14, 10, 15. Find the fifth.

3 marks4 minsaverages-and-range-q3

Show solution

Worked solution

1.Spot the skill: Mean = sum / count.
2.Use the find the mode stage first, then find the mean.
3.Keep the final answer visible: 13.

Final answer

Mark points

M1: use the correct mean = sum / count. median = middle value when ordered. mode = most frequent value.range = largest − smallest. the mean is sensitive to outliers; the median is more robust.use the appropriate average for the context.
A1: 13

Watch out

Hardproblem solving

Which average is most appropriate for shoe sizes? Justify.

3 marks5 minsaverages-and-range-q4

Show solution

Worked solution

1.Spot the skill: Mean = sum / count.
2.Use the find the mean stage first, then order the data and find the range.
3.Keep the final answer visible: Mode — it identifies the most common size.

Final answer

Mode — it identifies the most common size

Mark points

M1: use the correct mean = sum / count. median = middle value when ordered. mode = most frequent value.range = largest − smallest. the mean is sensitive to outliers; the median is more robust.use the appropriate average for the context.
A1: Mode — it identifies the most common size

Watch out

Exam-stylemulti-step

A dataset has values 2, 4, 6, 8, 100. Which average is most affected by the outlier?

4 marks6 minsaverages-and-range-q5

Show solution

Worked solution

1.Spot the skill: Mean = sum / count.
2.Use the order the data and find the range stage first, then find the median.
3.Keep the final answer visible: Mean.

Final answer

Mean

Mark points

M1: use the correct mean = sum / count. median = middle value when ordered. mode = most frequent value.range = largest − smallest. the mean is sensitive to outliers; the median is more robust.use the appropriate average for the context.
A1: Mean

Watch out

Grade 9 stretchproblem solving

Five numbers have mean 12. Four of the numbers are 8, 11, 13 and 15. Find the fifth number.

4 marks7 minsaverage-g9

Show solution

Worked solution

1.Use mean x count to find the total.
2.Subtract the four known numbers.

Final answer

Mark points

M1: total = 60
M1: known total = 47
A1: 13

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Averages and range - 2 marksFind the median of: 2, 5, 8, 12, 15, 19.Mark answer

Answer

10 (mean of 8 and 12)

2Collecting and sampling data - 2 marksWhy might a questionnaire question be biased?Mark answer

Answer

Leading wording, only offering responses that agree, or not including a 'no' option

3Grouped data and estimated mean - 2 marksWhich class contains the median for 30 values with frequencies 8, 12, 10?Mark answer

Answer

The 15th and 16th values — both in the second class [frequency 8, then 8+12=20]

4Charts, tables and diagrams - 3 marksA back-to-back stem-and-leaf shows boys' scores and girls' scores. How do you compare distributions?Mark answer

Answer

Compare medians and ranges for each group

Mastery check

I can explain the method for averages and range.
I can show clear working without skipping key steps.
I can avoid this mistake: Students find the median without sorting the data first, $pic$ king a middle-looking value from the unsorted list.Always sort. With an even number of values, the median is the mean of the two middle values.

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