11 plus Mean, Median, Mode & Range questions

What a Mean, Median, Mode & Range question looks like

A Mean, Median, Mode & Range question in the 11 plus gives your child a set of numbers and asks what the set as a whole does: its average, or how spread out it is. The numbers come as a short list in the question, as a table, or as a bar chart or pictogram. That last route is what separates this type from the paper’s other data questions: a bar chart or pictogram question is finished once a value has been read off the figure, whereas here reading it is only the first step.

Four things get asked for. The mean is the one most children meet first: add everything up, then share it out evenly. The middle value is what you get by putting the numbers in order and taking the one in the middle. The most common value is whichever number turns up most often. And the difference between the biggest and the smallest tells you how spread out the set is.

About half the questions ask for one of those four straight out. The rest run the other way: an average is given, one number is missing, and your child works back to it. The hardest few add a number to a set and move the average, combine two groups of different sizes, or give three numbers as X, 2X and 3X.

Example one: the mean of a short list

Five children scored these marks in a quiz: 6, 8, 8, 9, 14. What is the mean mark? The options are 8, 9, 10, 11 and 14.

Add the five marks. 6 and 8 is 14, plus 8 is 22, plus 9 is 31, plus 14 is 45. There are five marks, so share the 45 out between them: 45 divided by 5 is 9.

The answer is 9.

Two wrong options are worth naming. 8 tempts because it is the only mark that appears twice, so a child who has just finished a most-common question reaches for it. 11 comes from dividing 45 by 4 instead of 5, the commonest slip in the type: miscounting how many numbers there are.

Example two: the middle value from a table

A table shows how many steps Mia walked on each of seven days. When the seven numbers are put in order from smallest to largest, what is the middle value?

Put them in order before anything else: 4900, 5400, 6800, 7500, 8200, 8700, 9100. Seven numbers means the middle one is the fourth, with three each side. The fourth is 7500.

The answer is 7500.

The option that catches children is 6800. It is the fourth row down the table, so anyone who counts to the middle of the table instead of ordering the numbers first lands on it and feels confident. The middle row and the middle value only match if the table is already in order, and here it is not.

Example three: working backwards from an average

Five test scores have a mean of 78. Four of the scores are 65, 84, 79 and 72. What is the fifth score? The options are 75, 78, 82, 88 and 90.

The mean tells you the total. If five scores share out evenly to 78 each, all five together must come to 5 times 78, which is 390. Now add the four you have been given: 65 and 84 is 149, plus 79 is 228, plus 72 is 300. The fifth score is what is left: 390 take away 300, which is 90.

The answer is 90.

78 is the option built to tempt, because it is already sitting in the question. It would only be right if the missing score were exactly average, and four average scores would come to 312 while these four come to 300, so the missing one has to be well above 78. The other trap is 75, which is 300 divided by 4: the average of the scores you already have.

How to answer it

1. Read the question twice and decide which of the four it wants before touching a number. The words are the instruction: “mean” means add and share out, “middle value” means put them in order, “most common” means count how often each value turns up.
2. Write every number down in a row. If they are on a chart or in a table, copy them out before calculating: mixing reading with working out is where values go missing.
3. Count how many numbers there are and write that down too. For a mean it is what you divide by, and for a middle value it tells you which position to count to.
4. Do the one calculation the question asked for, and no others.
5. Check the answer is a sensible size. A mean and a middle value always land between the smallest and the largest number in the set, so anything outside that is wrong already.

Common mistakes

• Answering with the wrong one of the four. In the quiz-marks question above, 8 is the most common mark and 9 is the mean, and both are on offer. The wrong options are built for a child who has settled into one kind of question and keeps going.
• Finding the middle value without ordering the numbers. The numbers are put deliberately out of order, and whatever sits in the middle of the unsorted list is always one of the five options.
• Dividing by the wrong count. Dividing a total of 45 by 4 rather than 5, or forgetting that adding a number to a set changes the count as well as the total.
• Adding when the question wants a difference. With the list 12, 5, 8, 17, 9 the spread is 17 take away 5, which is 12. The option 22 is 17 plus 5.
• Assuming a missing value equals the average. When the mean is given and a number is missing, the mean itself is always one of the options and is almost never the answer.

How to practise 11 plus Mean, Median, Mode & Range questions

The useful practice here is not more arithmetic, it is more deciding. Give your child a set of numbers and ask which of the four the question wants and why, before they work anything out. That is the step the marks turn on. When one comes back wrong, ask which calculation they did rather than what answer they got. Short and regular beats long and occasional, which is the pattern we set out in how much 11 plus practice a child needs each day. 11+ Daily has a Mean, Median, Mode & Range project alongside the other data topics, on laptop, tablet or phone, and the full list sits on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed from another publisher, and the bank is re-checked in rounds.

Questions parents ask