The reverse question is the discriminator in 11+ averages: "the mean of five numbers is 12; four are given; find the fifth". Work out the total first (5 × 12 = 60), then subtract the numbers you already have. Master that one move and averages becomes one of the most reliable mark-scoring topics on the paper. This page teaches the mean, median, mode and range in full, shows every method worked step by step, and points you to free downloadable worksheets and timed mocks so your child can practise until the method is automatic.
What 11+ averages questions actually test
At 11+ level, "averages" means the three measures of average — mean, median and mode — plus the range, which measures spread. Examiners like this topic because it rewards careful, ordered working rather than clever tricks, yet it still separates children through reverse problems and multi-step data questions. Our free worksheet library grades every skill across four tiers — Foundation, Developing, Secure and Greater Depth — so a child can enter at their level and climb.
- Mean: add all the values, then divide by how many values there are.
- Median: the middle value once the numbers are in order.
- Mode: the value that appears most often (there can be more than one, or none).
- Range: the largest value minus the smallest value — a measure of spread, not an average.
Every method worked step by step
Take this data set: 7, 3, 9, 7, 4. Here is how each measure is found from the same five numbers.
- Mean: total = 7 + 3 + 9 + 7 + 4 = 30. There are 5 values, so mean = 30 ÷ 5 = 6.
- Median: reorder to 3, 4, 7, 7, 9. The middle (3rd of 5) value is 7.
- Mode: 7 appears twice, everything else once, so the mode is 7.
- Range: 9 − 3 = 6.
The median trap appears with an even count. For 3, 4, 7, 9 there is no single middle number, so take the two central values (4 and 7), add them and halve: (4 + 7) ÷ 2 = 5.5. Children who forget to reorder first, or who pick one middle number instead of averaging the pair, lose easy marks here.
The reverse-mean method (the discriminator)
Reverse-mean questions are where the marks are won and lost. The single rule to memorise is: total = mean × number of values. Work back from the total every time.
- Multiply the mean by the count to rebuild the total.
- Add up the values you already know.
- Subtract the known total from the full total to find the missing value.
Worked example: the mean of five numbers is 12. Four of them are 10, 15, 8 and 13. Full total = 12 × 5 = 60. Known total = 10 + 15 + 8 + 13 = 46. Missing value = 60 − 46 = 14. The same idea handles "a new value is added and the mean changes" questions — always rebuild the totals before and after.
Quick-reference formula box
| Measure | How to find it |
|---|---|
| Mean | total of values ÷ number of values |
| Median | middle value in order (average the two middle values if the count is even) |
| Mode | most frequent value |
| Range | largest value − smallest value |
| Missing value | (mean × count) − sum of known values |
Common mistakes and exam traps
- Not ordering before finding the median — the most frequent error by far. Always rewrite the list smallest to largest first.
- Confusing range with an average — the range is spread, not a central value. Read the question word carefully.
- Forgetting the even-count median rule — average the two middle numbers.
- Miscounting how many values there are in the mean — divide by the correct count, especially when zeros appear in the data.
- Assuming there is always exactly one mode — a set can be bimodal or have no mode at all.
- The last-sentence switch — a worded problem may give the mean and ask for the total, or vice versa. Underline exactly what is wanted.
How averages appear in a real GL 11+ paper
Most regions now sit GL Assessment papers after CEM's separate 11+ tests were phased out around 2022–2023. A GL maths paper is typically around 50 multiple-choice questions in about 50 minutes — roughly a minute per question — answered A–E on a separate OMR answer sheet, with no calculator. Averages usually appear as one or two questions, often bundled inside a bar chart or table so the child must read data and calculate. Essex CSSE and some independent papers use free-response answers and lean a little more on multi-step data and probability. Whichever your region, topic mastery is what lifts the raw score into a strong Standard Age Score (SAS, roughly 69–141, 100 = average, with many grammar schools needing around 111+). Practise under time with our free mock papers and region-matched centre-specific mocks.
Difficulty progression: Year 5 vs Year 6
Year 5 (easier): "Find the mean of 4, 6, 8 and 10." Add to 28, divide by 4, answer 7. Direct, one-step, all values given.
Year 6 (exam-level): "The mean age of four children is 9. A fifth child joins and the mean becomes 9.4. How old is the fifth child?" Rebuild totals: 4 × 9 = 36; 5 × 9.4 = 47; fifth child = 47 − 36 = 11. Same core rule, two totals, one extra step — exactly the jump examiners use to stretch the top tier.
How to use the free worksheets
- Diagnose: one Secure-tier sheet, unaided. Score under 70%? Drop a tier. Over 90%? Climb one.
- Little and often: 15–20 minutes daily beats a weekend marathon — retention lives in the gaps between sessions.
- Review errors out loud: the child explains what went wrong; you resist explaining first. Self-diagnosis sticks.
- Fold into mocks: once a topic hits Greater Depth, let full timed mock papers keep it warm rather than more isolated drilling.
Every averages worksheet — with answer keys and OMR sheets — is free in our worksheet library under the Maths tab. No card, no email wall. For format-specific practice, see our GL paper guide and revision hub.
Frequently asked questions
Is a calculator allowed for 11+ averages questions?
No. GL 11+ maths papers are non-calculator, so children must be fluent with mental and written addition and short division to find a mean quickly and accurately.
How many averages questions come up in the exam?
Usually one or two, often hidden inside a data or chart question. They carry as much weight as any other question, so reliable method matters more than the small number of them.
What is the single most important averages skill to practise?
The reverse-mean method — turning a mean back into a total to find a missing value. It is the most common discriminator and rewards children who always rebuild the total first.