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SATs practice · Year 6Chapter 8 · Statistics and data
Year 6 · Chapter 8

Statistics and data

By the end of this chapter you will be able to

  • Read a table or bar chart, and add or subtract the values to answer a question
  • Work out the angle or the number of people in a pie chart sector
  • Read a line graph and find the change between two readings
  • Find the mean, median, mode and range of a set of values
  • Compare two data sets, and say whether a conclusion from a graph is fair

You should already know

  • Reading bar charts and tables (Year 4 and Year 5 statistics)
  • Multiplication, division and fractions of amounts (Chapter 2 of this book)

Where this leads

  • Practice paper: reasoning (Chapter 9 of this book)

KS2 SATs practice for reasoning (Year 6). Original questions in the style of the national tests. Content: Solve comparison, sum and difference problems using information in graphs and tables (Year 5-6), Complete, read and interpret tables, including timetables (Year 5), Interpret and construct pie charts and line graphs, and use them to solve problems (Year 6), Calculate the mean as an average, and find a missing value when the mean is known (Year 6), Find the mode, median and range of a data set (Year 5-6), Compare two data sets and explain which is more suitable, and why (Year 6), Say whether a conclusion from a graph is valid, and explain (Year 6).

Exercise P  Do you remember?

Non-calculator
1
FWork out 14 + 9 + 7.
2
FWhich is the biggest number: 45, 54 or 39?
3
FWhat is 360 divided by 4?
4
FHow many is 25% of 80?
5
FWhich number comes next in the list 3, 6, 9, 12, __?
8.1

Tables, bar charts and comparisons

Year 6
Solve comparison, sum and difference problems using information in graphs and tables (Year 5-6), Complete, read and interpret tables, including timetables (Year 5)
Video lesson: Tables, bar charts and comparisonsComing soon
Practice material. Do this without a calculator. This is practice in the style of the Year 6 maths reasoning test. It is not the official test, and no official test item is used. Marks and timings are a practice guide only.
Story. Year 6 surveyed 64 pupils about how they travel to school. The results are in a table and a bar chart, and the class uses them to answer questions about the totals and the differences.

Rule

To answer a question from a table or chart, first find the values you need. Then add (for a total) or subtract (for a difference).

Read the labels carefully. Check what the question asks: "how many more" means subtract.

Key ideas

  • Read the labels and the values before you calculate.
  • Total = add the values. Difference = subtract the smaller from the larger.
  • Check the total of all the values before you answer.
How pupils travelNumber of pupils
Bus18
Walk25
Cycle9
Car12
Pupils and how they travel to school.
Worked example 8.1.1
Use the table. What is the total number of pupils in Bus and Walk?

Method. Read the values from the table, then add them.

  1. Find the values you need. Read the values for Bus and Walk from the table.
    18 + 25
  2. Add them together. Add the values.
    18 + 25 = 43
  3. Write the answer.
    43

Check. 43 is the sum of the chosen values. ✓

Worked example 8.1.2
Use the table. How many more pupils are there in Walk than in Car?

Method. Subtract the smaller value from the larger value.

  1. Read the two values. Walk has 25 and Car has 12.
    25 and 12
  2. Subtract the smaller from the bigger. Find how many more Walk there are than Car.
    25 - 12 = 13
  3. Write the answer.
    13

Check. 12 + 13 = 25. ✓

Worked example 8.1.3
Which way of travel is the most popular, according to the table? Write the name.

Method. The most popular is the biggest value.

  1. Find the biggest value. The biggest number is 25.
    25
  2. Name it. The label next to 25 is Walk.
    Walk
  3. Write the answer.
    Walk

Check. 25 is the largest value. ✓

Watch out

Watch out. "How many more" means subtract. "How many in total" means add.

Exam tip

Practice guide. Copy the values you need into a short list before you calculate.

Exercise 8A  Practice: tables and bar charts

Non-calculator
1
FUse the table. What is the total number of pupils in Cycle and Car?
How pupils travelNumber of pupils
Bus18
Walk25
Cycle9
Car12
2
FUse the table. What is the total number of pupils in Bus and Walk and Cycle?
18Bus25Walk9Cycle12Carpupils
3
FUse the table. How many more pupils are there in Walk than in Cycle?
18Bus25Walk9Cycle12Carpupils
4
FUse the table. How many more pupils are there in Bus than in Car?
18Bus25Walk9Cycle12Carpupils
5
PSUse the table. What is the total number of pupils in Bus and Walk and Cycle and Car?
6
FUse the table. What is the total number of pupils in Apples and Plums?
Fruit soldBags sold
Apples120
Pears85
Plums60
Grapes95
7
FUse the table. How many more bags are there in Grapes than in Plums?
Fruit soldBags sold
Apples120
Pears85
Plums60
Grapes95
8
PSUse the table. How many more bags are there in Apples than in Pears?
Fruit soldBags sold
Apples120
Pears85
Plums60
Grapes95
9
PSThe shop sold 360 bags of fruit in total, according to the table. How many bags were sold of apples and pears together?
Fruit soldBags sold
Apples120
Pears85
Plums60
Grapes95
10
FTick the statements that are true for the travel table. Options: More pupils walk than cycle | Car has more pupils than cycle | Bus has the fewest pupils | Walk and bus together have fewer than 40 pupils. Write the options you would tick, in the order given.
18Bus25Walk9Cycle12Carpupils
11
RWhich statement from the bar chart is valid? Choose one: Walking is the most common way to travel, Nobody cycles to school or More pupils came by bus than by car and cycle together.
18Bus25Walk9Cycle12Carpupils
12
RThe table says 64 pupils were surveyed. Explain how you would check that the numbers in the table add up to 64.
8.2

Pie charts

Year 6
Interpret and construct pie charts, and use them to solve problems (Year 6)
Video lesson: Pie chartsComing soon
Story. The school council asks 60 pupils for their favourite after-school club. The results are shown as a pie chart, and the council must calculate how many pupils chose each club from the angles of each sector.

Rule

A pie chart is a circle split into sectors. The whole circle is 360 degrees.

Angle of a sector = (number in the group / total) x 360. Number in the group = (angle / 360) x total.

Key ideas

  • The whole pie is 360 degrees.
  • Sector angle = share of the total x 360.
  • The sector angles add up to 360 degrees.
Worked example 8.2.1
A pie chart shows 72 pupils. 18 pupils chose Bus. What is the angle of the sector for Bus, in degrees?

Method. Find the fraction of the total, then multiply by 360.

  1. Share of the total. Bus is 18 out of 72.
    18 / 72
  2. Work out the angle. A whole pie is 360 degrees, so multiply the share by 360.
    18 / 72 x 360 = 90
  3. Write the answer.
    90

Check. 90 / 360 = 18 / 72. ✓

Watch out. Make sure all the sector angles add up to 360.

Worked example 8.2.2
A pie chart of 60 pupils has a sector of 120 degrees for Walk. How many pupils chose Walk?

Method. Find the fraction of the whole pie, then multiply by the total.

  1. Share of the pie. The sector is 120 of the 360 degrees.
    120 / 360
  2. Multiply by the total. Multiply the share by the total of 60.
    120 / 360 x 60 = 20
  3. Write the answer.
    20

Check. 20 / 60 = 120 / 360. ✓

Worked example 8.2.3
A pie chart of 40 pupils has a sector of 135 degrees for Chess. How many pupils chose Chess?

Method. Find the fraction of the whole pie, then multiply by the total.

  1. Share of the pie. The sector is 135 of the 360 degrees.
    135 / 360
  2. Multiply by the total. Multiply the share by the total of 40.
    135 / 360 x 40 = 15
  3. Write the answer.
    15

Check. 15 / 40 = 135 / 360. ✓

Watch out

Watch out. A sector of 180 degrees is half the pie, not a quarter. A quarter is 90 degrees.

Exam tip

Practice guide. Work out the fraction first, then multiply by 360 or by the total.

Exercise 8B  Practice: pie charts

Non-calculator
1
FA pie chart shows 60 pupils. 15 pupils chose Football. What is the angle of the sector for Football, in degrees?
2
FA pie chart shows 60 pupils. 10 pupils chose Swimming. What is the angle of the sector for Swimming, in degrees?
3
FA pie chart shows 36 pupils. 6 pupils chose Art. What is the angle of the sector for Art, in degrees?
4
FA pie chart of 40 pupils has a sector of 72 degrees for Netball. How many pupils chose Netball?
5
FA pie chart of 50 pupils has a sector of 180 degrees for Dance. How many pupils chose Dance?
6
PSA pie chart of 80 pupils has a sector of 45 degrees for Cricket. How many pupils chose Cricket?
7
PSA pie chart shows 72 pupils. The sectors for Chess, Choir and Drama are 120, 90 and 60 degrees. The rest of the pupils are in Gardening. How many pupils are in the Gardening sector?
8
FA sector of a pie chart is 180 degrees. What fraction of the pupils does it show? Choose one: a half, a quarter or a third.
9
RTick the statements that are true about pie charts. Options: The sector angles add up to 360 degrees | Each sector angle is a fraction of the total | A bigger sector always means fewer people | Each sector can be bigger than 360 degrees. Write the options you would tick, in the order given.
10
RExplain why a sector of 90 degrees must show a quarter of the pupils.
11
PSA pie chart shows 80 pupils. 20 pupils chose Music. What is the angle of the sector for Music, in degrees?
12
PSA pie chart of 90 pupils has a sector of 72 degrees for Choir. How many pupils chose Choir?
8.3

Line graphs and readings

Year 6
Interpret and construct line graphs, and use them to solve problems (Year 6)
Video lesson: Line graphs and readingsComing soon
Story. Ali records the temperature in the greenhouse every two hours. He plots the readings on a line graph to see how the temperature changes through the day.

Rule

A line graph joins readings taken over time. To find the change between two times, subtract the earlier reading from the later one.

The steepest part of the line shows the biggest change between two readings.

Key ideas

  • Read the value at each time from the graph or table.
  • Change = later reading - earlier reading.
  • A steep line means a big change over a short time.
TimeTemperature (degrees C)
6 am5
8 am9
10 am14
12 pm17
2 pm15
Greenhouse temperature readings.
Worked example 8.3.1
The table shows readings taken at different times. How much did the value change between 6 am and 12 pm, in degrees C?

Method. Subtract the earlier reading from the later one.

  1. Read the two values. At 6 am it is 5 degrees C, and at 12 pm it is 17 degrees C.
    5 and 17
  2. Find the change. Subtract the first reading from the second.
    17 - 5 = 12
  3. Write the answer.
    12 degrees C

Check. 5 + 12 = 17. ✓

Worked example 8.3.2
The table shows readings taken at different times. How much did the value change between Jan and May, in cm?

Method. Subtract the earlier reading from the later one.

  1. Read the two values. At Jan it is 20 cm, and at May it is 31 cm.
    20 and 31
  2. Find the change. Subtract the first reading from the second.
    31 - 20 = 11
  3. Write the answer.
    11 cm

Check. 20 + 11 = 31. ✓

Worked example 8.3.3
At which time was the temperature highest? Use the table.

Method. Look for the largest value.

  1. Find the largest value. The largest is 17 degrees.
    17
  2. Find its time. 17 degrees is at 12 pm.
    12 pm
  3. Write the answer.
    12 pm

Check. 17 is the highest reading. ✓

Watch out

Watch out. A line graph shows readings only at the times that were measured. Do not assume what happened between them.

Exam tip

Practice guide. Read each value with a ruler, and check the scale on the axis before you read it.

Exercise 8C  Practice: line graphs

Non-calculator
1
FThe table shows readings taken at different times. How much did the value change between 8 am and 2 pm, in degrees C?
TimeTemperature
6 am5
8 am9
10 am14
12 pm17
2 pm15
2
FThe table shows readings taken at different times. How much did the value change between Mar and Jul, in cm?
MonthHeight (cm)
Jan20
Mar26
May31
Jul29
3
FWhat was the temperature at 10 am? Write the number of degrees.
TimeTemperature
6 am5
8 am9
10 am14
12 pm17
2 pm15
4
PSBetween which two readings did the plant grow the most? Write the months.
MonthHeight (cm)
Jan20
Mar26
May31
Jul29
5
FBetween which two readings did the plant grow by 6 cm? Write the months.
MonthHeight (cm)
Jan20
Mar26
May31
Jul29
6
RA line graph shows a plant growing steadily. Can you say it will be 40 cm next month? Explain.
7
FThe temperature rose from 9 degrees at 8 am to 17 degrees at 12 pm. By how many degrees did it rise? Write the number.
TimeTemperature
6 am5
8 am9
10 am14
12 pm17
2 pm15
8
RWhich statement about the temperature graph is valid? Choose one: The temperature was highest at 12 pm, The temperature was 20 degrees at 2 pm or The temperature rose every time between readings.
9
RA line graph of a plant height has readings in May and July of 31 cm and 29 cm. Did the plant grow between May and July? Write yes or no.
10
PSThe table shows readings taken at different times. How much did the value change between 10 am and 2 pm, in degrees C?
TimeTemperature
6 am5
8 am9
10 am14
12 pm17
2 pm15
8.4

Mean, median, mode and range

Year 6
Calculate the mean as an average, and find a missing value when the mean is known (Year 6), Find the mode, median and range of a data set (Year 5-6)
Video lesson: Mean, median, mode and rangeComing soon
Story. Five children measure how far they can jump, in centimetres. The teacher wants the average, the middle value, the most common value and how spread out the results are.

Rule

The mean is the total divided by the number of values. The median is the middle value when they are in order. The mode is the most common value. The range is the largest minus the smallest.

Key ideas

  • Mean = total / number of values.
  • Median: order the values first; with an even number, take halfway between the middle two.
  • Range = largest - smallest.
Worked example 8.4.1
Lengths of five beans. The values are 4, 6, 8, 10, 12 cm. What is the mean, in cm?

Method. The mean is the total divided by the number of values.

  1. Add the values. Add up all 5 values.
    4 + 6 + 8 + 10 + 12 = 40
  2. Share equally. Divide the total by the number of values, 5.
    40 / 5 = 8
  3. Write the answer.
    8 cm

Check. 8 x 5 = 40. ✓

Watch out. Check that the total of the values is the mean times the number of values.

Worked example 8.4.2
Find the median of these values: 7, 3, 9, 5, 1.

Method. Order the values, then find the middle one.

  1. Put the values in order. Write them from smallest to largest.
    1, 3, 5, 7, 9
  2. Find the middle. With 5 values, the middle value is 5.
    5
  3. Write the answer.
    5

Check. There are 2 values on each side of 5. ✓

Watch out. Always order the values first. The median is not the first or the last value.

Worked example 8.4.3
What is the mode of these values: 2, 5, 3, 5, 8, 5, 2?

Method. Count how often each value appears, and choose the most common.

  1. Count how many times each value appears. 2 appears 2 time(s), 3 appears 1 time(s), 5 appears 3 time(s), 8 appears 1 time(s)
    5 appears 3 times
  2. Choose the most common. The mode is the value that appears most often.
    5
  3. Write the answer.
    5

Check. 5 appears 3 times, which is the most. ✓

Watch out

Watch out. The median is found after putting the values in order. The first value you see is not the median.

Exam tip

Practice guide. Write the values in order first, then the median, mode and range are easy to see.

Exercise 8D  Practice: mean, median, mode and range

Non-calculator
1
FFour strips of ribbon. The values are 3, 5, 7, 9 cm. What is the mean, in cm?
2
FSix bags of potatoes. The values are 10, 12, 14, 16, 18, 20 kg. What is the mean, in kg?
3
PSFive test scores. There are 5 values in all, with the values 8, 6, 10, 7 and one missing value. The mean is 8. What is the missing value?
4
PSFour scores. There are 4 values in all, with the values 10, 15, 11 and one missing value. The mean is 12. What is the missing value?
5
FFind the median of these values: 14, 3, 9, 11, 6, 2, 8.
6
PSFind the median of these values: 3, 8, 10, 13.
7
FWhat is the mode of these values: 4, 4, 7, 9, 9, 9, 2?
8
FWhat is the range of these values: 12, 4, 9, 20, 7?
9
FWhat is the range of these values: 25, 31, 18, 27, 22?
10
RWhich average is affected most by one very large value? Choose one: the mean, the median or the mode.
11
RExplain why the mean of 10 and 20 is 15, but the median is also 15.
12
PSFive children have a mean height of 140 cm. Four of them are 136, 142, 138 and 144 cm. How tall is the fifth child, in cm?
13
RThe median of a set of values is 9. There are seven values, and the middle one is missing. What is the middle value? Write the number.
14
FWhat is the mode of these values: 5, 8, 8, 2, 5, 9, 8?
15
PSWhat is the range of these values: 3, 17, 9, 12, 6, 15?
8.5

Comparing data sets and valid conclusions

Year 6
Compare two data sets and explain which is more suitable, and why (Year 6), Say whether a conclusion from a graph is valid, and explain (Year 6)
Video lesson: Comparing data sets and valid conclusionsComing soon
Story. Two football teams have scored the same total number of goals in five matches. Coach Ruth wants to know which team is more consistent, because her team plays best when it scores about the same every match.

Rule

Two data sets can have the same mean but very different spreads. The range shows how spread out the values are.

A conclusion is valid only if the data support it. A link in a graph does not prove that one thing causes another.

Key ideas

  • Same mean does not mean same spread: compare the range too.
  • A smaller range means more consistent results.
  • Check whether a conclusion goes beyond what the data show.
Worked example 8.5.1
Team A scored 5, 6, 7, 6, 6 goals. Team B scored 1, 6, 6, 6, 11 goals. Which team is more consistent? Write A or B.

Method. Compare the ranges, because a smaller range means more consistent scores.

  1. Range A. Largest 7, smallest 5, so the range is 2.
    7 - 5 = 2
  2. Range B. Largest 11, smallest 1, so the range is 10.
    11 - 1 = 10
  3. Compare. Team A has the smaller range, so its scores are more consistent.
    A
  4. Write the answer.
    A

Check. 2 < 10, and both teams have a mean of 6. ✓

Worked example 8.5.2
Ben says that because ice cream sales go up on hot days, ice cream makes the weather hotter. Is his conclusion valid? Write yes or no.

Method. A link between two things does not show that one causes the other.

  1. What the data show. Sales and hot days go up together.
    a link
  2. Is it a cause? Ice cream cannot make the weather hotter, so the conclusion does not follow.
    no
  3. Write the answer.
    no

Check. A link is not proof of cause. ✓

Worked example 8.5.3
Two classes have a mean score of 12. Class A has a range of 3 and Class B has a range of 15. Which class has more consistent scores? Write A or B.

Method. Compare the ranges: the smaller range means more consistent scores.

  1. Range of A. The range of Class A is 3.
    3
  2. Range of B. The range of Class B is 15.
    15
  3. Compare. A has the smaller range, so its scores are more consistent.
    A
  4. Write the answer.
    A

Check. 3 < 15. ✓

Watch out

Watch out. Two data sets with the same mean can be very different. Always check the range or the spread.

Exam tip

Practice guide. Write one sentence that compares the averages and one that compares the spread.

Exercise 8E  Practice: comparing data sets and conclusions

Non-calculator
1
RTeam X scored 4, 5, 6, 5, 5 goals. Team Y scored 2, 5, 5, 5, 8 goals. Which team is more consistent? Write X or Y.
2
RAli and Sam both have a mean score of 12 in a quiz. Ali has a range of 3, and Sam has a range of 15. Who had more consistent scores? Write Ali or Sam.
3
RTwo shops sell the same shoes. One shop says its average shoe size is 6, so most people buy size 6. Explain why this might not be true.
4
RA graph shows that the number of umbrellas sold goes up when it rains. Which conclusion is valid? Choose one: Umbrellas sold go up when it rains, Rain is caused by umbrella sales or Umbrella sellers make it rain.
5
RA bar chart shows the average for two classes. Class 1 has a mean of 20 and Class 2 has a mean of 24. Is Class 2 better at the test? Write yes or no.
6
PSFour journeys to school. The values are 12, 14, 16, 18 minutes. What is the mean, in minutes?
7
PSFind the median of these values: 2, 9, 4, 12, 7.
8
RExplain why the mode can be a useful average for shoe sizes.

Chapter summary

SkillMethod
Totals and differencesRead the values from a table or chart; add for totals, subtract for differences.
Pie chartThe whole is 360 degrees. Angle = share x 360; number = angle / 360 x total.
Line graph changeChange = later reading - earlier reading.
MeanTotal of the values divided by the number of values.
MedianThe middle value after ordering; for two middle values, halfway between them.
ModeThe value that appears most often.
RangeLargest value minus smallest value; smaller range means more consistent.
Valid conclusionOnly claim what the data show; a link does not prove a cause.

Can you…?

  • I can read a table or bar chart and add or subtract the values
  • I can work out the angle or the number of pupils in a pie chart sector
  • I can read a line graph and find the change between two readings
  • I can find the mean, median, mode and range of a data set
  • I can find a missing value when the mean is known
  • I can compare two data sets and say whether a conclusion is valid

Key terms

mean
the total of the values divided by how many values there are
median
the middle value when the values are in order
mode
the value that appears most often
range
the largest value minus the smallest value
sector
a slice of a pie chart
valid conclusion
a conclusion that the data really support

Exercise 8Z  Check-up

Non-calculator
1
Use the table. What is the total number of pupils in Bus and Cycle?
How pupils travelNumber of pupils
Bus18
Walk25
Cycle9
Car12
2
Use the table. How many more pupils are there in Walk than in Bus?
How pupils travelNumber of pupils
Bus18
Walk25
Cycle9
Car12
3
A pie chart shows 60 pupils. 24 pupils chose Dance. What is the angle of the sector for Dance, in degrees?
4
A pie chart of 36 pupils has a sector of 90 degrees for Netball. How many pupils chose Netball?
5
The table shows readings taken at different times. How much did the value change between 6 am and 2 pm, in degrees C?
TimeTemperature
6 am5
8 am9
10 am14
12 pm17
2 pm15
6
Four plants. The values are 6, 8, 10, 12 cm. What is the mean, in cm?
7
Four numbers. There are 4 values in all, with the values 4, 9, 7 and one missing value. The mean is 8. What is the missing value?
8
Find the median of these values: 11, 4, 8, 15, 2.
9
What is the mode of these values: 3, 7, 3, 9, 7, 3?
10
What is the range of these values: 14, 6, 22, 9?

Total 17 marks. Allow about 17 minutes.

Exercise 8X  Practice test questions

Non-calculator
1
FUse the table. What is the total number of pupils in Walk and Car?[1 mark]
How pupils travelNumber of pupils
Bus18
Walk25
Cycle9
Car12
2
FUse the table. How many more bags are there in Apples than in Plums?[1 mark]
Fruit soldBags sold
Apples120
Pears85
Plums60
Grapes95
3
FA pie chart shows 90 pupils. 30 pupils chose Reading. What is the angle of the sector for Reading, in degrees?[2 marks]
4
PSA pie chart of 36 pupils has a sector of 80 degrees for Chess. How many pupils chose Chess?[2 marks]
5
PSFour scores. There are 4 values in all, with the values 5, 9, 6 and one missing value. The mean is 7. What is the missing value?[2 marks]
6
FFind the median of these values: 12, 5, 9, 20, 3.[1 mark]
7
FWhat is the range of these values: 8, 15, 3, 11?[1 mark]
8
PSThe table shows readings taken at different times. How much did the value change between Jan and Jul, in cm?[2 marks]
MonthHeight (cm)
Jan20
Mar26
May31
Jul29
9
RTeam P has a mean score of 10 and a range of 2. Team Q has a mean of 10 and a range of 12. Which team is more consistent? Explain.[3 marks]
10
RBen says that because the mean of his scores is 9, most of his scores are 9. Is he right? Write yes or no.[2 marks]
← Chapter 7: Algebra and sequencesChapter 9: Practice paper: reasoning →

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