Averages — mean, median, mode and range — appear in nearly every 11+ maths paper, and they are some of the most reliable marks on the whole test. A child who muddles which is which throws away easy points; a child who knows the four methods cold can bank 5-10 marks here in a couple of minutes. This guide gives you the plain-English definitions, a full worked example of all four from one data set, the trickier "reverse mean" questions that separate strong candidates, and the exact traps that cost marks in a real GL paper.
The four definitions, in plain English
Three of these words are called averages (mean, median and mode) because each is a single number that stands in for a whole set. Range is not an average at all — it measures how spread out the numbers are — but examiners always test it alongside the other three, so learn all four together.
- Mean — add every value, then divide by how many values there are. This is the "average" most adults think of. For 5, 7 and 9: (5+7+9) ÷ 3 = 7.
- Median — the middle value once the numbers are put in order. Sort first, every time.
- Mode — the value that appears most often. A set can have two modes (bimodal) or no mode at all if nothing repeats.
- Range — the largest value minus the smallest value. It is a measure of spread, not a middle.
A memory hook that sticks: mode = most (both start with "mo"); median = middle (both start with "m-i-d"); mean is the "mean" one because it makes you do the most work.
All four from one data set (worked example)
Find the mean, median, mode and range of: 4, 7, 2, 7, 9.
- Sort into order first: 2, 4, 7, 7, 9. Do this before anything else — it makes the median and mode obvious and stops silly errors.
- Mean: (2 + 4 + 7 + 7 + 9) ÷ 5 = 29 ÷ 5 = 5.8.
- Median: five values, so the middle one is the third: 7.
- Mode: the number that appears most is 7 (it appears twice; every other value appears once).
- Range: highest minus lowest = 9 − 2 = 7.
Notice that the mean, median, mode and range can all turn out to be different numbers (5.8, 7, 7 and 7 here). Do not assume that because you found "7" once, it is the answer to every part.
Median when there is an even number of values
If the list has an even count there is no single middle number, so you take the mean of the two middle values. For 3, 5, 8, 12 (already sorted), the two middle numbers are 5 and 8, so the median is (5 + 8) ÷ 2 = 6.5. A median does not have to be one of the numbers in the list — a common surprise for children the first time they meet it.
Reverse-mean questions (the 11+ favourite)
The single most common "harder" averages question works the mean backwards: you are told the mean and asked to find a missing value. The method never changes — turn the mean back into a total using total = mean × number of values, then compare with what you already have.
- "The mean of 5 numbers is 12. Four of them are 10, 11, 13 and 15. Find the fifth." The total must be 12 × 5 = 60. The four known numbers add to 49. So the fifth is 60 − 49 = 11.
- "What number added to 3, 7 and 4 makes the mean 5?" After adding one number there will be four values, so the total needed is 5 × 4 = 20. The three given numbers total 14, so the missing number is 20 − 14 = 6.
- "A cricketer's mean over 6 innings is 30. After a 7th innings the mean rises to 32. How many did he score?" Old total = 30 × 6 = 180; new total = 32 × 7 = 224; the innings scored 224 − 180 = 44.
Once a child sees that "mean" and "total" are two views of the same thing, this entire question family becomes routine.
Quick-reference formula box
| Measure | How to work it out | What it tells you |
|---|---|---|
| Mean | total of values ÷ number of values | the balancing "average" |
| Median | middle value of the sorted list (mean of the two middle if even) | the typical middle |
| Mode | value that occurs most often | the most common result |
| Range | highest value − lowest value | the spread |
| Missing value | total = mean × number of values, then subtract the known part | reverse-mean problems |
How this appears in the exam
In a GL Assessment maths paper — now the dominant format across most UK 11+ regions since the standalone CEM tests were phased out around 2022-2023 — averages usually show up as a two- or three-part question inside a roughly 50-questions-in-50-minutes paper (about a minute a question), multiple-choice with answers A-E marked on a separate OMR sheet, and no calculator. Because the options are printed, you can sometimes work backwards from them, but it is quicker to calculate and then find your answer in the list. Averages appear across Year 5 and Year 6 practice; the mean and range come first (Year 4/5 level), while reverse-mean and even-count medians are the Year 6 exam-level twists. CSSE (the Essex consortium) and many independent-school papers include free-response versions where you must show the working, so practise writing the total-and-divide line out, not just picking a letter.
Exam traps that cost real marks
- Forgetting to sort before reading off the median or mode — the number-one error. Always reorder the list first.
- Confusing mode and median — "most" versus "middle". Say the memory hook in your head each time.
- Dropping a value from the mean — miscount the numbers and your division is wrong. Count them twice.
- Even-count median — remembering to average the two middle values instead of guessing one.
- Mode of zero or two — if nothing repeats there is no mode; if two values tie, there are two. Don't force a single answer.
- The last-line switch — a worded problem that gives you a data set then asks for the range when you were expecting the mean. Underline the actual question before you calculate.
Practise it the fast way
Averages reward speed built on accuracy, so drilling short sets against the clock pays off. Quest Arena's free maths worksheets include focused averages drill sheets, and you can pressure-test the skill under exam timing with our free mock papers and region-specific centre mocks. Parents building a term-by-term plan can map the whole maths syllabus with the revision hub, check the timeline on our key dates page, and see exactly what a GL paper looks like on the GL papers guide. Everything is free to use.
Frequently asked questions
Is a calculator allowed for averages questions?
No. GL 11+ maths papers are non-calculator throughout, so children need confident mental and column arithmetic. The numbers in averages questions are deliberately kept manageable, but a division like 29 ÷ 5 = 5.8 still has to be done by hand — practising short division without a calculator is essential.
What is the difference between mean, median and mode?
The mean shares the total out evenly (add everything, divide by how many). The median is the middle value once the numbers are in order. The mode is the value that appears most often. Range is the odd one out — it is not an average, it is the highest minus the lowest and measures spread.
Which year should my child master averages?
Basic mean and range are secure Year 4/5 skills, and the exam-level twists — reverse-mean problems and medians of even-length lists — should be solid by autumn of Year 6 when 11+ testing begins. Little-and-often timed practice works far better than one long session.