GCSE Maths · Higher · Chapter 3: Statistical diagrams and averages

Charts, averages from tables and scatter graphs

Grades 4–9About 85 minutesCalculator allowed84 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Work out the mean of 4, 7, 9, 12 and 13.
[2 marks]
Answer

A2
Work out the median of 8, 3, 11, 6, 9 and 14.
[2 marks]
Answer
A3
Write down the mode and the range of 5, 9, 2, 9, 7 and 4.
[2 marks]
Answermode $\dots\dots$ range $\dots\dots$
A4
A pie chart is drawn for 72 people. What angle stands for one person?
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

30 students said how many pets they own: 6 own none, 11 own 1, 8 own 2 and 5 own 3. Work out the mean number of pets.

Hint Multiply each number of pets by its frequency and add the results. Divide by the total frequency, not by the number of rows.

B1
The table shows the number of goals scored in 25 football matches. Work out the mean number of goals.
[3 marks]
Answer

Example 2

30 people waited for a bus. 5 waited up to 10 minutes, 12 waited more than 10 and up to 20 minutes, 8 waited more than 20 and up to 30 minutes, and 5 waited more than 30 and up to 40 minutes. Work out an estimate of the mean waiting time.

Hint You do not know the exact times, so use the midpoint of each class. That is why the answer is only an estimate.

B2
The table shows the masses, $m$ kg, of 30 parcels. Work out an estimate of the mean mass.
[3 marks]
Answer

Example 3

120 students chose their favourite sport: 45 chose football, 25 chose tennis, 30 chose swimming and 20 chose another sport. Work out the angle for each sector of a pie chart.

Hint The whole circle is

36 0^{\circ}

for 120 students, so each student gets

360 \div 120 = 3^{\circ}

B3
90 people said how they travel to work: 36 by bus, 27 by car, 18 walk and 9 cycle. Work out the angle for each sector of a pie chart.
[2 marks]
Answer

Example 4

8 students recorded the hours they spent revising and their test marks: (1, 30), (2, 38), (3, 41), (4, 50), (5, 52), (6, 61), (7, 63) and (8, 72). Plot the points on a scatter graph and describe the correlation. Then draw a line of best fit and use it to estimate the mark of a student who revised for 4.5 hours.

Hint A line of best fit follows the trend with about as many points above it as below it. Read up from 4.5 on the horizontal axis to the line, then across.

B4
The scatter graph shows the maximum temperature on 10 days and the number of hot drinks a café sold.
(a)
Describe the correlation.
[1 mark]
Answer
(b)
Draw a line of best fit. Use it to estimate the number of hot drinks sold on a day with a maximum temperature of 15 °C.
[2 marks]
Answer
Total for B4: 3 marks

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The stem-and-leaf diagram shows the ages of 15 people at a meeting.
(a)
Find the median age.
[1 mark]
Answer
(b)
Find the range of the ages.
[1 mark]
Answer
(c)
What fraction of the people are older than 40?
[1 mark]
Answer
Total for C1: 3 marks

C2
The pie chart shows how 180 people travel to work. How many of them travel by train?
[2 marks]
Answer
C3
The bar chart shows the number of books read by a group of boys and a group of girls in three months.
(a)
In which month did the boys read more books than the girls?
[1 mark]
Answer
(b)
Work out the total number of books the girls read in the three months.
[1 mark]
Answer
Total for C3: 2 marks
C4
The table shows the shoe sizes of 30 students.
(a)
Write down the mode.
[1 mark]
Answer
(b)
Work out the median.
[2 marks]
Answer
Total for C4: 3 marks
C5
The table shows the heights, $h$ cm, of 30 plants. Write down the modal class and the class that contains the median.
[2 marks]
Answermodal class $\dots\dots\dots$ median class $\dots\dots\dots$
C6
The scatter graph shows the age and the value of 9 cars of the same model.
(a)
Describe the correlation.
[1 mark]
Answer
(b)
One car does not fit the pattern. Write down its age and its value.
[1 mark]
Answer
Total for C6: 2 marks
C7
The salaries of five people who work in a small office are £18 000, £19 000, £21 000, £22 000 and £95 000.
Which average gives the best idea of a typical salary? Give a reason, and work out that average.
[2 marks]
Answer
C8
Five whole numbers have a mean of 8. Four of the numbers are 3, 7, 9 and 13. Find the fifth number.
[2 marks]
Answer
C9
The pictogram shows the number of cars that used a car park on three days.
Work out the total number of cars on the three days.
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The table shows the number of children in some families. The mean number of children is 1.5.
Work out the value of $x$ .
[4 marks]
Answer $x = \dots\dots$

D2
Sam says, "More students walk to school in school B, because the walking sector in its pie chart is bigger."
School A has 600 students. In its pie chart, the sector for walking is $7 2^{\circ}$ . School B has 300 students. In its pie chart, the sector for walking is $12 0^{\circ}$ .
Explain why Sam is wrong.
[3 marks]
D3
The table shows the distances, $d$ km, that 60 people travel to work.
(a)
Work out an estimate of the mean distance. Give your answer to 3 significant figures.
[3 marks]
Answer
(b)
Explain why your answer to part (a) is only an estimate.
[1 mark]
Answer
Total for D3: 4 marks
D4
The scatter graph shows the midday temperature on 10 days and the number of ice creams a shop sold.
(a)
Describe the relationship between the temperature and the number of ice creams sold.
[1 mark]
Answer
(b)
Draw a line of best fit. Use it to estimate the number of ice creams sold on a day with a midday temperature of 22 °C.
[2 marks]
Answer
(c)
Explain why it would not be sensible to use your line to estimate the sales on a day at 40 °C.
[1 mark]
Answer
Total for D4: 4 marks
D5
15 students in class A took a test. Their marks are shown in the stem-and-leaf diagram.
In class B, the median mark was 62 and the range was 28.
(a)
Find the median mark for class A.
[2 marks]
Answer
(b)
Find the range of the marks for class A.
[1 mark]
Answer
(c)
Compare the marks of the two classes.
[2 marks]
(d)
What fraction of class A scored more than 60?
[1 mark]
Answer
Total for D5: 6 marks
D6
The mean height of the 12 players in a basketball squad is 1.82 m. A new player joins the squad. The mean height of the 13 players is now 1.83 m.
Work out the height of the new player.
[3 marks]
Answer
D7
A pie chart shows the favourite type of music of a group of people. The sector for rock music has an angle of $8 1^{\circ}$ and stands for 27 people.
(a)
How many people are in the group?
[2 marks]
Answer
(b)
38 people chose pop music. Work out the angle of the pop music sector.
[1 mark]
Answer
Total for D7: 3 marks
D8
Five positive whole numbers have a mean of 12, a median of 13, a mode of 15 and a range of 9.
Find the five numbers. You must show your working.
[4 marks]
Answer
D9
The table shows the ages, $a$ years, of 40 people at a party. Estimate how many of the people are older than 35.
[2 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Seven positive whole numbers have a mode of 3, a median of 5 and a mean of 6. There is only one mode.
Find the largest possible range of the seven numbers.
[4 marks]
Hint 1 · What to try
Write the seven numbers in order. The total is $7 \times 6 = 42$ . To make the range large, make the smallest number small and the largest number big.
Hint 2 · The first line
3 must appear more often than any other number, so it appears at least twice, below the median. The smallest number can then be 1: $1, 3, 3, 5, \dots$
Hint 3 · The full method
The numbers above the median must all be different, so they are at least 6 and 7. Then $1 + 3 + 3 + 5 + 6 + 7 + g = 42$ gives $g = 17$ . The range is $17 - 1 = 16$ .
Answer

E2
There are 30 students in a class. The mean mark of the boys in a test is 60. The mean mark of the girls is 72. The mean mark of the whole class is 67.2.
How many girls are in the class?
[3 marks]
Hint 1 · What to try
Call the number of girls $g$ . Then there are $30 - g$ boys.
Hint 2 · The first line
The total of all the marks is $30 \times 67.2 = 2016$ , and it is also $60 (30 - g) + 72 g$ .
Hint 3 · The full method
$1800 + 12 g = 2016$ , so $12 g = 216$ and $g = 18$ .
Answer
E3
In a pie chart, the angle for apples is $2 0^{\circ}$ more than the angle for bananas. 9 more people chose apples than chose bananas.
How many people are shown in the pie chart?
[3 marks]
Hint 1 · What to try
The difference in the angles stands for the difference in the numbers of people.
Hint 2 · The first line
$2 0^{\circ}$ stands for 9 people.
Hint 3 · The full method
The whole circle is $36 0^{\circ}$ , which is $18 \times 2 0^{\circ}$ . So the number of people is $18 \times 9 = 162$ .
Answer
E4
Five points have $x$ -values 2, 3, 5, 6 and 9, and $y$ -values 12, 17, 26, 30 and 45. A line of best fit has a gradient of 4 and passes through the point whose coordinates are the mean of the $x$ -values and the mean of the $y$ -values.
Use the line to estimate the value of $y$ when $x = 7$ .
[3 marks]
Hint 1 · What to try
Work out the mean of the $x$ -values and the mean of the $y$ -values.
Hint 2 · The first line
The means are $5$ and $26$ , so the line passes through $(5, 26)$ .
Hint 3 · The full method
From $x = 5$ to $x = 7$ is 2 across, so the line goes up $2 \times 4 = 8$ . The estimate is $26 + 8 = 34$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw and read pictograms, bar charts, pie charts and stem-and-leaf diagrams.
I can work out the mean, median, mode and range from a frequency table.
I can estimate the mean from grouped data and find the modal class.
I can describe correlation, draw a line of best fit and say when an estimate from it can be trusted.

Answers and mark scheme

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