Average questions in the 11 plus: why you cannot average two averages

Averages look like the friendliest topic in 11 plus maths. Add them up, share them out, done. It is also the topic where a child can do correct arithmetic, show sensible working, and hand in a wrong answer three questions in a row for the same reason.

The reason is not that the sums are hard. It is that an average is a summary, and you cannot do arithmetic on a summary.

Here is the question that proves it.

A swimming club has two groups. The 10 children in the junior group swim a mean of 12 lengths each. The 30 children in the senior group swim a mean of 20 lengths each. What is the mean number of lengths for the whole club?

The answer is 18.

The answer most often given is 16.

Where 16 comes from, and why it is not carelessness

There are two averages in that question and nothing else that looks like a number to work with. A child who has been taught that an average is the middle of things looks at 12 and 20, finds the middle, and writes 16.

That is real arithmetic, done correctly. It is also the answer to a question nobody asked, because it quietly assumes the two groups are the same size.

They are not. There are three times as many seniors as juniors, so the club's average has to sit much closer to 20 than to 12. Sixteen sits exactly halfway, which is the one place it cannot be.

The step that goes missing is the one nobody says out loud: you cannot combine two averages. You have to go back to the totals.

Ten juniors swimming 12 each is 120 lengths. Thirty seniors swimming 20 each is 600 lengths. Together that is 720 lengths swum by 40 children, and 720 shared by 40 is 18.

Eighteen is nearer 20 than 12, which is exactly what you would expect once you can see the 600 against the 120. The bigger group pulls the answer towards itself, and an average on its own has no way of telling you that, because the size of the group is the thing an average throws away.

The same move, in a question that looks nothing like it

Freya's mean score across 5 spelling tests is 14. After her sixth test, her mean is 15. What did she score in the sixth test?

This one usually gets 15, and sometimes 14.5.

Both come from the same place as the 16. A child reaches for the means and does something reasonable with them, because the means are what is written down.

Go back to the totals and there is almost nothing to it. Five tests with a mean of 14 is 70 marks. Six tests with a mean of 15 is 90 marks. She scored 20.

Twenty is a long way above 15, and that surprises most children. It has to be: one score has to drag the average of six things up by a whole mark on its own, so it must be well clear of the new average. A child who answers 15 has not just made a slip. They have missed the only interesting thing in the question.

And once more, in the gentlest version

Five numbers have a mean of 8. Four of them are 2, 5, 12 and 14. What is the fifth number?

The answer given is very often 8.

Again, it is not a guess. The mean is the number on the page, the question asks for a number, and 8 is a number that belongs to the list in some sense. It just is not this one.

Five numbers with a mean of 8 have a total of 40. The four we were given come to 33. The fifth is 7.

Three questions, one move

Those three look like three different topics. A tutor would file them as combining groups, a changed mean, and a missing value. For the child they are the same question, and they are fixed by the same sentence:

Turn every average back into a total before you do anything with it.

Totals can be added, taken away and compared. Averages cannot. Once a child has that, all three questions above become ordinary arithmetic, and the reason this is worth drilling is that it converts a whole family of hard questions into easy ones in a single move.

It is also why "do more averages practice" so often changes nothing. A child who happily finds the mean of a list will keep finding the mean of a list, beautifully, and go on losing every question where two averages have to meet.

How to tell which problem your child has

Ask them the swimming club question. The answer tells you where they are without a mark sheet.

• 18, with 120 and 600 written down somewhere. The move is there.
• 18, with no working at all. Worth asking how they got it, because 18 can also be a lucky guess from "it should be nearer 20".
• 16. The arithmetic is fine. The idea that an average hides the size of the group has not landed yet, and that is one conversation, not a term of work.
• Anything else. The mean itself is the thing to go back to first, before any of this.

Where it turns up when it does not say "averages" at the top

This is worth knowing, because a child can be solid in the averages chapter and still lose these:

• A batting or bowling average after one more game
• A class test where one absent child sits the paper later
• Two shops, two delivery times, and a question about both
• A mean worked out from a frequency table, where each value happens several times
• Any question where somebody joins a group or leaves it

None of those pages says "you cannot average two averages". Every one of them fails in exactly the same place, on the total that has to be rebuilt before anything else can happen.

Try it on one paper

Our free paper marks itself the moment your child finishes. You will see which skills are solid, which are not, and the named mistake behind each one that went wrong. It costs nothing and there is no card.