GCSE Maths · Higher · Chapter 18: Sampling and more complex diagrams

Sampling, frequency polygons, cumulative frequency, box plots and histograms

Grades 4–9About 85 minutesCalculator allowed90 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Write down the midpoint of the class $25 < t \leq 40$ .
[1 mark]
Answer

A2
Write 54 out of 360 as a fraction in its simplest form.
[1 mark]
Answer
A3
Find the median of $11, 4, 9, 15, 7, 12$ .
[2 marks]
Answer
A4
A bar is 6 units wide and has an area of 42 square units. Work out its height.
[1 mark]
Answer
A5
The frequencies of four classes are 5, 12, 9 and 4. Write down the running totals.
[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

A school has 1200 students. 270 of them are in Year 7. A stratified sample of 80 students is taken by year group. How many Year 7 students are in the sample?

Hint Year 7 is

\frac{270}{1200}

of the school, so it is the same fraction of the sample.

B1
A club has 900 members. 315 of them are juniors. A stratified sample of 60 members is taken. How many juniors are in the sample?
[2 marks]
Answer

Example 2

On a histogram, the bar for the class $20 < t \leq 45$ has a frequency density of 1.6. How many people are in this class?

Hint Frequency is frequency density times class width. The width here is

45 - 20

B2
On a histogram, the bar for the class $30 < m \leq 50$ has a frequency density of 2.4. How many people are in this class?
[2 marks]
Answer

Example 3

Find the median and the interquartile range of these 11 values.
3, 5, 6, 8, 9, 11, 12, 14, 15, 18, 21

Hint With 11 values the median is the 6th, the lower quartile the 3rd and the upper quartile the 9th.

B3
Find the median and the interquartile range of these 15 values.
21, 24, 25, 27, 30, 31, 33, 34, 36, 38, 40, 41, 45, 47, 52
[3 marks]
Answermedian $= \dots\dots$ , IQR $= \dots\dots$

Example 4

The times of 30 runners are grouped: 4 in $0 < t \leq 10$ , 9 in $10 < t \leq 20$ , 12 in $20 < t \leq 30$ and 5 in $30 < t \leq 40$ . Work out an estimate of the mean time.

Hint Use the midpoint of each class for every runner in it. Multiply, add, then divide by 30.

B4
The waiting times of 40 people are grouped: 6 in $0 < m \leq 20$ , 13 in $20 < m \leq 40$ , 16 in $40 < m \leq 60$ and 5 in $60 < m \leq 80$ . Work out an estimate of the mean waiting time.
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The table shows the number of students in each year group at a school. A stratified sample of 60 students is taken by year group. How many Year 8 students should be in the sample?
[2 marks]
Answer

C2
Ella wants to find out how often the students in her school use the library. She asks 30 students who are in the library at lunchtime. Give two reasons why her sample may be biased.
[2 marks]
C3
The frequency polygon shows the heights of a group of students.
(a)
How many students are in the group?
[1 mark]
Answer
(b)
Write down the modal class.
[1 mark]
Answer
(c)
Work out an estimate of the mean height.
[3 marks]
Answer
Total for C3: 5 marks
C4
The cumulative frequency graph shows the times taken by 80 people to finish a puzzle.
(a)
Use the graph to find the median time.
[1 mark]
Answer
(b)
Use the graph to find the interquartile range.
[2 marks]
Answer
(c)
How many people took more than 30 minutes but no more than 60 minutes?
[2 marks]
Answer
Total for C4: 5 marks
C5
The box plots show the test scores of a Year 9 group and a Year 11 group.
(a)
Write down the median score of the Year 11 group.
[1 mark]
Answer
(b)
Work out the interquartile range of each group.
[2 marks]
AnswerYear 9 $\dots\dots$ Year 11 $\dots\dots$
(c)
Compare the scores of the two groups.
[2 marks]
Total for C5: 5 marks
C6
The histogram shows the masses of some apples.
(a)
How many apples have a mass of more than 40 grams?
[2 marks]
Answer
(b)
Work out an estimate of the number of apples with a mass between 25 grams and 50 grams.
[3 marks]
Answer
Total for C6: 5 marks
C7
The table shows the times some students took to solve a puzzle. Write down the coordinates of the points you would plot to draw a frequency polygon.
[2 marks]
Answer
C8
The table shows the heights of some plants. Which class contains the median height?
[2 marks]
Answer
C9
A gym has 800 members. Describe how to take a simple random sample of 50 members.
[2 marks]

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 students in Year 10 and Year 11 at a school. A sample of 60 students is taken, stratified by year group and by gender. Work out the number of Year 11 girls in the sample.
[3 marks]
Answer

D2
Jo says, "More people took between 10 and 15 minutes than in any other class, because that bar is the tallest."
The histogram shows the times some people took to get to work. Is Jo right? You must show your working.
[3 marks]
Answer
D3
On a histogram, the bar for the class $10 < x \leq 20$ has a height of 3.6 and stands for 36 people. The class $20 < x \leq 50$ has 54 people. Work out the height of its bar.
[2 marks]
Answer
D4
The histogram shows the ages of the people at a concert. Some of the frequencies are missing from the table.
(a)
Use the histogram to work out the three missing frequencies in the table.
[2 marks]
Answer
(b)
Work out an estimate of the number of people aged between 15 and 50.
[2 marks]
Answer
(c)
Work out an estimate of the median age.
[2 marks]
Answer
Total for D4: 6 marks
D5
The cumulative frequency graph shows the masses of 120 tomatoes. The lightest tomato has a mass of 104 grams and the heaviest has a mass of 212 grams.
(a)
Use the graph to find the interquartile range.
[2 marks]
Answer
(b)
Draw a box plot for this information. Write the five values you use.
[2 marks]
Answer
Total for D5: 4 marks
D6
Two groups of seedlings were given different feeds. The box plots show their heights after four weeks. Compare the heights of the two groups.
[2 marks]
D7
A company has 2400 workers. 1680 work in an office and the rest work in a factory. Ana takes a stratified sample of 50 workers.
(a)
How many factory workers should be in her sample?
[2 marks]
Answer
(b)
Explain why a stratified sample is better than asking the first 50 workers who arrive in the morning.
[1 mark]
Total for D7: 3 marks
D8
The histogram shows the times some customers waited at a counter. Work out an estimate of the mean waiting time. Give your answer to 1 decimal place.
[4 marks]
Answer
D9
The table shows the masses of some dogs. Work out an estimate of the number of dogs with a mass of more than 35 kg.
[2 marks]
Answer
D10
In a stratified sample of 50 students from a school, 14 were in Year 10. There are 182 Year 10 students in the school. Work out the number of students in the school.
[3 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
The histogram shows the times some people took to answer a quiz question. There is no scale on the frequency density axis. 18 people took between 10 and 20 seconds. How many people are shown on the histogram?
[4 marks]
Hint 1 · What to try
Count small squares. The $10 < t \leq 20$ bar stands for 18 people, so work out how many people one small square stands for.
Hint 2 · The first line
Find the area of every bar in small squares, then compare each with the $10 < t \leq 20$ bar.
Hint 3 · The full method
The bars have frequencies in the ratio $12 : 18 : 24 : 20 : 6$ , so the total is $12 + 18 + 24 + 20 + 6 = 80$ .
Answer

E2
Some people were timed. 5 took $0 < t \leq 10$ minutes, $x$ took $10 < t \leq 20$ , 12 took $20 < t \leq 30$ and 3 took $30 < t \leq 40$ . The estimate of the mean time is 20 minutes. Work out the value of $x$ .
[4 marks]
Hint 1 · What to try
Write the estimate of the mean as a fraction in terms of $x$ , using the midpoints.
Hint 2 · The first line
$\frac{25 + 15 x + 300 + 105}{20 + x} = 20$
Hint 3 · The full method
$430 + 15 x = 400 + 20 x$ , so $5 x = 30$ and $x = 6$ .
Answer $x = \dots\dots$
E3
Seven whole numbers, in order, are
$2, a, 6, 8, b, 15, c$
The interquartile range is 10, the range is 16 and the mean is 9. Find $a$ , $b$ and $c$ .
[4 marks]
Hint 1 · What to try
With seven values the lower quartile is the 2nd and the upper quartile is the 6th.
Hint 2 · The first line
$15 - a = 10$ and $c - 2 = 16$ .
Hint 3 · The full method
$a = 5$ and $c = 18$ , then $2 + 5 + 6 + 8 + b + 15 + 18 = 63$ , so $b = 9$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can take a simple random or a stratified sample, and say why a sample may be biased.
I can draw and read a frequency polygon, and estimate a mean from grouped data.
I can read the median and quartiles off a cumulative frequency graph, draw a box plot and compare two box plots.
I can work out and use frequency density, and estimate from a histogram.

Answers and mark scheme

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