Data and averages
Mean, median, mode and range, bar charts, pie charts, pictograms, tables and chance: what each question asks, the quick way to answer it, and the traps the wrong answers set.
Chapter 1
What are data questions?
You get some numbers, as a list, a chart, a table or a diagram, and one question about them. Most ask for an average or a reading off a chart. A few ask about chance.
The numbers. A short list, like this one, or a picture to read them from.
The question word. Mode, median, mean and range are four different jobs, so read the word twice.
Only the mean is on the primary curriculum, in Year 6. Mode, median, range and chance are usually taught at secondary school, but some 11+ papers ask them, so they’re all here.
What is the median?
13, 16, 2, 4, 10
The options
- Option 1: 2
- Option 2: 10
- Option 3: 4
- Option 4: 13
A Maths question from Rush, level 2.
Four words, one list: 2, 9, 4, 2, 8
-
Mode: the most common
The number that comes up most often.
2 comes up twice, so the mode is 2. Mode and most both start with mo. Trap: 9 is the biggest, not the most common.
-
Median: the middle
The middle number, once the list is in order.
In order the list is 2, 2, 4, 8, 9, so the median is 4. Chapter 2 shows how.
-
- 2 + 9 + 4 + 2 + 8 = 25
- 25 ÷ 5 = 5
Mean: share it out
Add them all up, then share by how many there are.
25 shared between 5 numbers is 5 each. Trap: 25 is the total. A mean is never bigger than the biggest number.
-
9 − 2 = 7
Range: the spread
The biggest take away the smallest.
9 − 2 = 7, wherever they sit in the list. Trap: 8 − 2, the last number take away the first.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
What is the median?
13, 16, 2, 4, 10
The options
- Option 1: 2
- Option 2: 10
- Option 3: 4
- Option 4: 13
A Maths question from Rush, level 2.
-
Step 1
Put the list in order.
Write the numbers smallest to biggest: 2, 4, 10, 13, 16. Count them as you go: five in, five out, so none got lost.
-
Step 2
Cross off one from each end.
Cross off the smallest and the biggest together: 2 and 16. Then 4 and 13. One number is left: 10.
Two pairs crossed off: 10 is left.
-
Step 3
Check it’s in the middle.
10 has two numbers below it, 2 and 4, and two above it, 13 and 16. The same on both sides, so 10 is the median.
Two mistakes it sets
| The mistake | Do this instead |
|---|---|
| Taking the middle of the list as written. 2 sits in the middle of 13, 16, 2, 4, 10. But in order, it’s the smallest. | Put the list in order first, every time. |
| Crossing off unevenly. 4 and 13 are one place out. 4 has one number below it and three above. | Cross off in pairs, one from each end, then count both sides. |
So it’s 10, option 2. Two numbers below it and two above, once the list is in order.
Remember: Order it. Cross off the ends.
Chapter 3
Everything else Rush asks.
Each card is one kind of question: the rule, one example, and the trap. On a chart, the first job is always the same: work out what each line is worth.
-
- 3 × 8 = 24
- 7 + 4 = 11
- 24 − 11 = 13
A number missing from the mean
Mean × how many = the total.
The mean of 3 numbers is 8, and two of them are 7 and 4. All three make 3 × 8 = 24, so the third is 24 − 11 = 13. Trap: 24 and 11 are steps on the way, not the answer.
-
Pictograms: part of a symbol
Read the key, then work out the part.
Each square is 8 books, so 7 squares are 56 and half a square is 4: 60 books. Three quarters of a square worth 12 is 9. Trap: 56, which leaves the half out.
-
Bar charts: read the scale first
First, find what one line is worth.
These go up in 20s, so a faint line halfway is 10 more. Mint stops halfway between 40 and 60: 50. How many more mint than lemon? 50 − 30 = 20. Trap: 80, from adding.
-
Mode and range on a chart
Mode: the tallest bar. Range: tallest take away shortest.
The mode is dog: the answer is the bar’s name, not 14. The range is 14 − 4 = 10. Trap: 14 is only the tallest. The shortest still comes off.
-
Line graphs: up, then across
Go up from the day to the point, then across to the scale.
When was it warmest? At the highest point: Wednesday, 12°C. How much warmer than Friday? 12 − 4 = 8°C. Trap: reading the point next door.
-
Conversion graphs
Up from one axis to the line, then across to the other.
About how many km is 50 miles? Up from 50 to the line, then across: 80 km. Each axis has its own scale: 25s along, 40s up. Trap: 50, the number you started with.
-
Pie charts: what fraction?
Turn the slice into a fraction of the total.
64 children chose a fruit. Grapes is an eighth: 64 ÷ 8 = 8. Grapes or apples is five eighths: 5 × 8 = 40. A square corner means a quarter. Trap: 16, the plums quarter.
-
Tickets sold Friday Saturday Total adult 24 ? 60 child 18 27 45 Total 42 63 105 Tables: where the row meets the column
Find the row, find the column, and read where they meet.
A ? in a table with totals is what’s left: 63 on Saturday take away 27 child tickets leaves 36 adult. Trap: 90, from adding. Bus timetables work the same way: the bus’s column, the stop’s row.
-
Venn diagrams: one circle at a time
Test the number against each circle on its own.
Where does 36 go? A multiple of 5? No. A multiple of 4? Yes, 4 × 9. So it goes in B only. Trap: forgetting the two other places, the overlap (in both) and outside (in neither).
-
Carroll diagrams: column, then row
One rule picks the column, the other picks the row.
Where does 77 go? It’s a multiple of 7 (7 × 11), so the first column. It’s odd, so the not even row. That’s box C. Trap: answering from one rule only.
Chance: out of everything
Chance is how many ways you want, out of everything that could happen. A bag has 5 red and 7 yellow counters, 12 in all, so the chance of red is 5/12. Trap: 5/7, which is red against yellow only.
In words: none is impossible, less than half is unlikely, exactly half is an even chance, more than half is likely, and all of them is certain.
Chapter 4: In the exam
The mean of a chart.
Some papers ask for the mean of a chart’s bars. Rush asks for the mean of a list, but it’s the same add and share, with one catch.
The questionThe chart shows the goals a team scored in 5 matches. What is the mean number of goals per match?
-
Step 1
Read every bar.
The scale goes up in 2s, so the faint lines are the odd numbers. The bars are 4, 0, 7, 3 and 6. Match 2 has no bar: no goals.
-
Step 2
Add them up.
4 + 0 + 7 + 3 + 6 = 20. The team scored 20 goals in all.
4 + 0 + 7 + 3 + 6 = 20
-
Step 3
Share by how many matches.
There were 5 matches, not 4 bars, so 20 ÷ 5 = 4. A match with no goals still counts.
20 ÷ 5 = 4
Two mistakes it sets
| The mistake | Do this instead |
|---|---|
| Sharing by the bars. Only 4 bars are drawn, and 20 ÷ 4 = 5. But there were 5 matches. | Count the matches along the bottom, not the bars. |
| Giving the total. 20 is all the goals added up. | Share it out. A mean can’t be bigger than the tallest bar, 7. |
So it’s 4 goals. 20 goals shared over 5 matches, counting the match with none.
Remember: Count the matches, not the bars.
Other ways the exam asks it
- A pie chart in degrees. A whole circle is 360°, so put the slice over 360. A 90° slice is a quarter. If 30 children were asked, a 60° slice is a sixth of them: 5.
- A median with two middle numbers. With an even count, it’s halfway between the middle two. For 3, 5, 8, 10 that’s halfway between 5 and 8: 6.5.
- “Which statement is NOT true?” Check each statement against the chart, one at a time. Underline NOT before you start.
- “When did it rise the most?” On a line graph, find the steepest piece of line: the biggest jump between two points next to each other.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Put the list in order.
- Cross off one from each end.
- Check it’s in the middle.
Remember: Order it. Cross off the ends.
For parents.
More on data and averages, and what parents ask us.
More on data and averages
Longer paper questions
Papers often ask several questions about one chart or table, and some need readings from several bars added up. Rush asks one question per picture and puts at most two readings together, such as a difference between two bars or two pie slices added. Scatter graphs, which ask for a judgement about a trend, and probability with two events, such as two coins or two dice, aren't in Rush, so practise them on paper.
Questions parents ask
How can my child remember mean, median, mode and range?
Mode is the most: both start with 'mo'. Median sounds like 'medium', the middle size. Range is how spread out the numbers are. Mean is the one that has to be worked out, by adding and sharing.
What if there are two middle numbers?
With an even number of values, the median is halfway between the two in the middle: for 3, 5, 8, 10 it's 6.5. Rush's median questions always give five numbers, so there is one middle number, but papers can set either.
Can the mean be a number that isn't in the list?
Yes, and it usually is. The mean of 5, 15 and 13 is 11, which isn't one of them. That's why the mean can't be spotted by looking: it has to be worked out.
How is probability asked at 11+?
In words (impossible, unlikely, even chance, likely, certain) or as a fraction: the number of ways to get what you want over the number of things that could happen. An even number on a 1 to 6 dice is 3 out of 6, an even chance.
Rush: 11+ Practice & KS2 SATsPractise data and averages in Rush.
- Data & Chance as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
- Data & Chance is in Premium, and six other games are free
On the App Store for iPhone and iPad
Six other games are free, plus a new family game every day. Data & Chance is in Premium: all 41 games are £24.99 a year after a free week1, or £3.99 a month.