Averages: The Three Slips, and How to Check

21 September 2026 · by Larry

Ask your child what the average of 6, 8 and 10 is and she will tell you 8 without pausing. Then she meets a question where the average is given and the amounts are not, and everything stops.

That is the shape of this topic. The calculating is easy and children get it quickly. The marks are lost somewhere else entirely — in what an average actually is, and in which direction the question is asking you to travel.

What an average really means

Most children learn a rule: add them up, divide by how many. The rule is correct and it is not the problem. The problem is that a rule is all they have.

An average is what everybody would get if you shared it out equally. Three children have 6, 8 and 10 sweets. Pour all 24 into one pile, share them out fairly, and each child gets 8. That is the average. Nothing has been lost and nothing invented — the same sweets, arranged evenly.

That picture is worth five minutes at the kitchen table, because every slip below comes from not having it.

The fact that unlocks the whole topic

If the average is the fair share, then putting the shares back together gives you the total:

average × how many = total

Children are taught the division. Very few are taught the multiplication, and almost every question that costs marks at P5 and P6 needs it. Nearly all of them give you an average and expect you to turn it back into a total before you can do anything useful.

The check: "The average of 4 numbers is 12. What do they add up to?" If she says 48 straight away, she has the idea. If she hesitates, this is the gap, and it is worth fixing before any harder question is attempted.

Slip one: averaging the averages

This is the one that catches the strongest children, because the wrong answer looks so reasonable.

Class A has 30 pupils, with an average mark of 80. Class B has 20 pupils, with an average mark of 70. What is the average mark of all 50 pupils?

Almost everyone says 75. It is the average of 80 and 70, it is tidy, and it is wrong.

It is wrong because there are more pupils in Class A, so Class A pulls harder. You cannot average two averages unless the groups are the same size.

Go back to totals, which is what the multiplication above is for. Class A's marks add up to 30 × 80 = 2400. Class B's add up to 20 × 70 = 1400. Altogether that is 3800 marks shared among 50 pupils, so the average is 3800 ÷ 50 = 76.

Seventy-six, not seventy-five. And notice it sits nearer to 80 than to 70, which makes sense once you see that the bigger class is doing more of the pulling.

The check: whenever two averages appear, ask "are the groups the same size?" If they are not, the answer is not halfway.

Slip two: working backwards

This is the one that costs the most marks in P6, and it is the same shape as the discount questions in ratio and percentage.

Ali's average over 3 tests was 80. After his 4th test, his average went up to 82. What did he score in the 4th test?

The instinct is to do something with 80 and 82 — to find the difference, or to add 2, and answer 84. Neither works, because an average is not a thing you can add to.

Turn both averages into totals and the question almost answers itself:

• After 3 tests: 3 × 80 = 240 marks.
• After 4 tests: 4 × 82 = 328 marks.
• The 4th test must have been 328 − 240 = 88.

He needed 88, not 84, because the new test has to lift its own weight and pull up the three that came before it. That is genuinely surprising to most children, and it is worth letting them be surprised by it.

The check: if a question gives you an average and asks for one missing amount, write both totals first. Every time.

Slip three: expecting the average to be one of the numbers

A smaller slip, but it makes children distrust a correct answer and go back and change it.

The average of 2, 2 and 8 is 4. There is no 4 anywhere in that list, and that is fine. An average does not have to be one of the numbers, and it often is not.

It can also be a decimal that is impossible in real life. The average number of children per family might be 2.4, and no family anywhere has 0.4 of a child. The average is not describing a family; it is describing how things would fall if they were shared evenly.

The check that does work: an average must land somewhere between the smallest number and the largest. If she works out an average of 95 from a set where the highest mark is 90, something has gone wrong. This is a quick sanity check she can use in an exam without redoing the sum, in the same family as the other habits that stop careless marks going.

The one habit that fixes all three

Write the total.

Before doing anything else, take every average in the question and turn it into a total. Write it down next to the working. Then look at the question again.

Three of these slips disappear the moment the totals are on the paper, because once you are working in totals you are just adding and subtracting amounts — which she has been able to do since P3. The difficulty was never the arithmetic. It was that averages are relationships rather than amounts, and relationships cannot be added together.

If you want to practise this

Do not start with exam questions. Start at the table with a handful of coins or sweets in unequal piles, and share them out. Let her see that the average is the flat, even version of what was already there.

Then ask the backwards question out loud: "if each of you ended up with 5, and there are 4 of you, how many did we start with?" That is the whole topic, and she will get it in a minute with objects in front of her when a worksheet might take a week.

After that, the written questions stop being a different subject.