11+ Averages: Mean, Median, Mode and Missing Numbers
Compare mean, median, mode and range, recover missing values and avoid unequal-group traps with eight original worked questions.
A collection of numbers can have several useful summaries. The important question is not just “Can I calculate an average?” but “Which summary has been requested, and what does it tell me?” Start by writing the name of the measure next to your working.
These original explanations and questions support preparation for 11-plus exams taken in 2027. They are not an official paper or a promise that every school tests these topics. Use your target school’s current guidance, and try each question before opening its worked answer.
Four measures, four different jobs
| Measure | Method | Check |
|---|---|---|
| Mean | Add the values and divide by their count | Include zero values in the count |
| Median | Order the values and locate the middle | For an even count, average the middle two |
| Mode | Find the most frequent value | There may be no mode or more than one |
| Range | Largest value minus smallest | This measures spread, not an average |
For 2, 4, 4, 7 and 13, the total is 30 and the count is five, so the mean is 6. The median is 4, the mode is 4 and the range is 11. The mean need not be one of the original values. A large value can pull it away from the middle observation; that does not make the calculation wrong.
Turn a mean back into a total
If six scores have mean 9, their total is 54. If five known scores add to 43, the missing score is 54 − 43 = 11. Multiply before subtracting: the given mean describes the whole group, not just the known members. Check by replacing the gap, adding all six scores and dividing by six.
Do not average unequal groups blindly
A group of three children collects a mean of 8 tokens each, while a group of five collects a mean of 12 each. Their totals are 24 and 60. Together there are eight children with 84 tokens, so the overall mean is 10.5. Simply averaging 8 and 12 gives 10, which gives the smaller group too much influence. Reconstruct totals whenever the group sizes differ.
Median questions need ordering even when the original list looks nearly sorted. Keep repeated values and zeros: deleting either changes the data. For an even number of observations, identify both central positions before doing the final average. For a mode question, count frequencies rather than selecting the largest number.
Eight questions to diagnose the method
Find the mean of 6, 8, 10 and 12.
Show answer and explanation
9
The total is 36 and there are four values. Calculate 36 ÷ 4 = 9.
Find the median of 11, 3, 7, 9 and 5.
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7
Order them: 3, 5, 7, 9, 11. The third value is 7.
Find the median of 2, 8, 4 and 10.
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6
Order them as 2, 4, 8, 10. The middle two are 4 and 8; their mean is 6.
Find the mode of 3, 5, 3, 8, 5 and 3.
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3
Three occurs three times; five occurs twice and eight once. The most frequent value is 3.
Find the range of 14, 6, 9 and 20.
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14
Subtract the smallest from the largest: 20 − 6 = 14. Do not subtract in list order.
Five scores have mean 12. Four scores are 9, 11, 13 and 15. Find the fifth.
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12
Required total is 5 × 12 = 60. Known total is 48. The missing score is 12.
The mean of 0, 7, 8 and 9 is required. What is it?
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6
Zero contributes nothing to the sum but still counts as one value: 24 ÷ 4 = 6.
Two players have mean score 6. Three others have mean score 11. Find the mean of all five scores.
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9
The totals are 12 and 33. Combined mean is 45 ÷ 5 = 9, not the mean of the two group means.
Explain the correction
If a child gets the number right but cannot describe the method, ask them to make a new small data set with the same mean. Then ask whether its median must also be the same. This separates understanding of a summary from remembering a sequence of arithmetic operations.
Return to the Maths learning hub, choose original practice and print resources, or continue with algebra basics. When reviewing, record the step you missed rather than just copying the answer. Explain one corrected question aloud, then try a different example of the same skill.
Teaching context checked 25 October 2026: DfE maths programme. This is curriculum context, not a school-specific exam specification. The examples, questions and explanations here are original; no official mark scheme or qualified-educator review is claimed.
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Choose original practice and use the answer explanations to decide what to work on next. These are not official school papers.