For: Year 5 and Year 6 | Exams: KS2 SATs and 11+ | Calculator: not allowed
Jump to: Level 1 – SATs core · Level 2 – SATs stretch · Level 3 – 11+ standard · Level 4 – 11+ selective
📌 KEY FACTS TO KNOW BY HEART
Charts: check the scale on a bar chart or line graph and the key on a pictogram before you read anything.
Averages: mean = total ÷ how many · median = middle value when in order · mode = most common · range = largest − smallest.
Pie charts: the whole circle is 360°. Angle for one item = 360° ÷ total. 180° = half, 90° = quarter, 120° = third.
Total from the mean: total = mean × how many.
Level 1 – SATs core: tables, pictograms and bar charts
Matches the chart and table questions in KS2 SATs Papers 2 and 3. Always check the scale or the key first.
✏️ WORKED EXAMPLE: The pictogram shows how many books four classes read. How many more books did Class C read than Class B?
Step 1: read the key: one circle = 10 books, so half a circle = 5 books.
Step 2: Class C has 5 circles: 5 × 10 = 50 books.
Step 3: Class B has 3½ circles: 30 + 5 = 35 books.
Step 4: 50 − 35 = 15 books
Common mistake: counting circles instead of books and answering 1½, or counting a half circle as a whole one.
Your turn
Q1.1 to Q1.3 use this bar chart.
Q1.1 How many children chose banana? Use the bar chart above.
Show answer
The banana bar reaches 12: 12 children
Q1.2 How many more children chose grapes than orange?
Show answer
Grapes 9, orange 5. 9 − 5 = 4 more
Q1.3 How many children took part in the survey altogether?
Show answer
8 + 12 + 5 + 9 + 3 = 37 children
Q1.4 to Q1.7 use this pictogram.
Q1.4 How many cakes were sold on Wednesday? Use the pictogram above.
Show answer
4 circles × 4 = 16 cakes
Q1.5 How many cakes were sold on Tuesday?
Show answer
2 circles = 8, and half a circle = 2. 8 + 2 = 10 cakes
Q1.6 How many more cakes were sold on Friday than on Thursday?
Show answer
Friday: 5 × 4 = 20. Thursday: 4 + 2 = 6. 20 − 6 = 14 more
Q1.7 How many cakes were sold altogether from Monday to Friday?
Show answer
12 + 10 + 16 + 6 + 20 = 64 cakes
Q1.8 to Q1.10 use this table of midday temperatures in five cities on one day.
| City | Midday temperature (°C) |
|---|---|
| London | 14 |
| Leeds | 11 |
| Cardiff | 15 |
| Glasgow | 9 |
| Belfast | 12 |
Q1.8 Which city was the warmest? Use the table above.
Show answer
The highest number is 15: Cardiff
Q1.9 What is the difference between the warmest and the coldest temperature?
Show answer
Warmest 15°C (Cardiff), coldest 9°C (Glasgow). 15 − 9 = 6°C
Q1.10 How many of the cities had a temperature higher than 11°C?
Show answer
London 14, Cardiff 15 and Belfast 12 are higher than 11. Leeds is exactly 11, so it does not count: 3 cities
Level 2 – SATs stretch: line graphs, two-way tables and the mean
Matches the harder reasoning questions in SATs Papers 2 and 3. Read across and down carefully.
✏️ WORKED EXAMPLE: Find the mean of 4, 7, 9, 5 and 10
Step 1: add the numbers: 4 + 7 + 9 + 5 + 10 = 35
Step 2: count them: there are 5 numbers.
Step 3: divide the total by how many there are: 35 ÷ 5 = 7
Common mistake: dividing by the wrong number (for example by 4), or picking the middle number instead of sharing the total equally.
Your turn
Q2.1 to Q2.4 use this line graph. The temperature in a greenhouse was measured every two hours.
Q2.1 What was the temperature at 12:00? Use the line graph above.
Show answer
The point above 12:00 is at 14°C
Q2.2 Between which two readings did the temperature rise the most?
Show answer
Rises: 3°C, 5°C, 3°C. The biggest is from 9°C to 14°C: 10:00 to 12:00
Q2.3 By how much did the temperature fall from 14:00 to 18:00?
Show answer
17°C at 14:00 and 11°C at 18:00. 17 − 11 = 6°C
Q2.4 What is the difference between the highest and lowest temperature shown?
Show answer
Highest 17°C, lowest 6°C. 17 − 6 = 11°C
Q2.5 to Q2.7 use this two-way table. It shows how some children travel to school.
| Walk | Bus | Car | |
|---|---|---|---|
| Boys | 8 | 5 | 3 |
| Girls | 6 | 7 | 4 |
Q2.5 How many girls travel by bus? Use the table above.
Show answer
Read across the Girls row to the Bus column: 7
Q2.6 How many children walk to school altogether?
Show answer
Boys 8 + girls 6 = 14 children
Q2.7 How many children are in the table altogether?
Show answer
Boys: 8 + 5 + 3 = 16. Girls: 6 + 7 + 4 = 17. Total: 33 children
Q2.8 Find the mean of 3, 8, 6 and 7
Show answer
Total 24, and there are 4 numbers. 24 ÷ 4 = 6
Q2.9 Four plants are 15 cm, 18 cm, 21 cm and 22 cm tall. What is their mean height?
Show answer
Total 76 cm. 76 ÷ 4 = 19 cm
Q2.10 The mean of five numbers is 9. What is the total of the five numbers?
Show answer
Total = mean × how many: 9 × 5 = 45
Level 3 – 11+ standard: pie charts, averages and missing values
Typical of grammar-school 11+ papers. A whole pie chart is 360°.
✏️ WORKED EXAMPLE: The pie chart shows the favourite colours of 60 children. How many children chose blue?
Step 1: the whole circle is 360° for 60 children.
Step 2: each child is 360 ÷ 60 = 6°
Step 3: blue is 150°, so 150 ÷ 6 = 25 children
Common mistake: reading the angle as the number of children and answering 150.
Your turn
Q3.1 to Q3.4 use this pie chart. It shows the favourite sports of 120 people.
Q3.1 What fraction of the people chose football? Give your answer in its simplest form. Use the pie chart above.
Show answer
120° out of 360° = 1/3
Q3.2 What angle should the cricket sector be?
Show answer
360 − 120 − 90 − 60 − 45 = 45°
Q3.3 How many people chose tennis?
Show answer
Each person is 360 ÷ 120 = 3°. 90 ÷ 3 = 30 people
Q3.4 How many more people chose football than swimming?
Show answer
Football 120 ÷ 3 = 40. Swimming 60 ÷ 3 = 20. 40 − 20 = 20 more
Q3.5 Find the median of 7, 3, 9, 4, 6, 10, 5
Show answer
In order: 3, 4, 5, 6, 7, 9, 10. The middle (4th) number is 6
Q3.6 The mean of four numbers is 8. Three of the numbers are 5, 9 and 11. What is the fourth number?
Show answer
Total of all four: 8 × 4 = 32. 32 − (5 + 9 + 11) = 32 − 25 = 7
Q3.7 to Q3.9 use this bar chart. 20 children were asked how many people live in their family (including themselves).
Q3.7 What is the mode of the family sizes? Use the bar chart above.
Show answer
The tallest bar is 3 people: 3
Q3.8 What is the median family size?
Show answer
With 20 children the median is between the 10th and 11th. Sizes in order: one 2, seven 3s (2nd to 8th), six 4s (9th to 14th). The 10th and 11th are both 4: 4
Q3.9 What is the mean family size?
Show answer
Total people: 2×1 + 3×7 + 4×6 + 5×4 + 6×2 = 2 + 21 + 24 + 20 + 12 = 79. 79 ÷ 20 = 3.95
Q3.10 The bar chart below shows how many books were borrowed from a class library from Monday to Thursday. The bar for Friday is missing. The mean number of books borrowed per day over the five days was 7. How many books were borrowed on Friday?
Show answer
Total for 5 days: 7 × 5 = 35. Monday to Thursday: 6 + 9 + 4 + 8 = 27. Friday: 35 − 27 = 8 books
Level 4 – 11+ selective: reasoning with averages and charts
The style of selective independent-school papers. Work with totals, not just averages.
✏️ WORKED EXAMPLE: The mean of six numbers is 10. A seventh number is added and the mean of all seven becomes 11. What is the seventh number?
Step 1: total of the six numbers: 6 × 10 = 60
Step 2: total of the seven numbers: 7 × 11 = 77
Step 3: the new number is the difference: 77 − 60 = 17
Check: (60 + 17) ÷ 7 = 11 ✓
Common mistake: thinking the new number must be 11 or 1. To raise the mean by 1 for all seven numbers, the new number has to be 7 more than the old mean.
Your turn
Q4.1 The mean age of five children is 9. Their teacher, who is 33, joins them. What is the mean age of all six people now?
Show answer
Children’s total: 5 × 9 = 45. New total: 45 + 33 = 78. 78 ÷ 6 = 13
Q4.2 Class A has 20 pupils with a mean score of 15. Class B has 10 pupils with a mean score of 18. What is the mean score of all 30 pupils?
Show answer
Totals: 20 × 15 = 300 and 10 × 18 = 180. 480 ÷ 30 = 16. (Not 16.5: the classes are different sizes.)
Q4.3 The mean of three numbers is 12. Two of the numbers are each increased by 6. What is the new mean?
Show answer
Old total 36. The total goes up by 12 to 48. 48 ÷ 3 = 16
Q4.4 to Q4.6 use these two pie charts. They show how pupils travel to two schools.
Q4.4 How many more pupils walk to School B than to School A? Use the pie charts above.
Show answer
School A: 90/360 = 1/4 of 240 = 60. School B: 60/360 = 1/6 of 480 = 80. 80 − 60 = 20 more
Q4.5 Both car sectors are 120°. How many more pupils travel by car to School B than to School A?
Show answer
120° is 1/3 of each chart. School A: 240 ÷ 3 = 80. School B: 480 ÷ 3 = 160. 160 − 80 = 80 more
Q4.6 How many pupils from the two schools together travel by bus?
Show answer
School A: 150/360 of 240 = 100. School B: 180/360 = 1/2 of 480 = 240. 100 + 240 = 340 pupils
Q4.7 The mean of eight numbers is 15. One number is removed and the mean of the seven numbers left is 14. Which number was removed?
Show answer
Old total 8 × 15 = 120. New total 7 × 14 = 98. 120 − 98 = 22
Q4.8 In a test, the 10 boys in a class had a mean of 12 marks. The whole class of 25 pupils had a mean of 15 marks. What was the mean mark of the girls?
Show answer
Class total 25 × 15 = 375. Boys’ total 10 × 12 = 120. Girls: 375 − 120 = 255 marks for 15 girls. 255 ÷ 15 = 17
Q4.9 90 children voted for a class pet and the results are shown on a pie chart. The dog sector is 120° and the fish sector is 60°. The rest voted for a rabbit or a hamster, and twice as many voted for a rabbit as for a hamster. How many children voted for a rabbit?
Show answer
Each child is 360 ÷ 90 = 4°. Rabbit and hamster share 360 − 120 − 60 = 180°, in the ratio 2 : 1, so rabbit is 120°. 120 ÷ 4 = 30 children
Q4.10 Five whole numbers have a mode of 4, a median of 6 and a mean of 7. The largest number is 12. What are the five numbers?
Show answer
The median means the middle number is 6. The mode 4 means at least two 4s, and they must be the two smallest. Total = 5 × 7 = 35, so the missing number is 35 − 4 − 4 − 6 − 12 = 9. The numbers are 4, 4, 6, 9, 12