What is S1: Sampling?

A population is the whole group being studied; a sample is the smaller group from which data are collected.

How many marks are S1 questions worth?

In verified past-paper appearances, S1 questions on Edexcel GCSE Maths 1MA1 have been worth 2–4 marks.

Which past exam papers has S1 appeared in?

S1 has appeared in 9 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Infer properties of populations or distributions from a sample, while knowing the limitations of sampling

Sampling

Worked answers, methods and verified real exam appearances for S1 on Edexcel GCSE Maths 1MA1.

Explanation

A population is the whole group being studied; a sample is the smaller group from which data are collected.
If the sample is representative, its proportion or mean can be used to estimate a population value.
Scale a sample proportion by the population size when estimating a count.
The result is an estimate, not a certainty.
Small samples, convenience sampling, under-coverage and non-response can make the sample unrepresentative and the inference unreliable.

Worked example

In a random sample of $75$ residents, $27$ support a proposal. Estimate how many of the town's $1250$ residents support it and state the assumption needed.

1.Sample proportion $=\dfrac{27}{75}=0.36$ .
2.Estimated population count $=0.36\times1250=450$ .
3.The estimate assumes that the sample is representative of the town's residents.

Answer: About $450$ residents, provided the sample is representative.

Common mistakes

Don't fall into the trap of using the sample frequency as the population estimate without scaling.
Don't fall into the trap of assuming a large but biased sample must be representative.

Exam tip

State a limitation in context, explaining how it could make the sample unrepresentative.

Worked practice

Tier 1 · Easy

In a random sample of

50

library users,

30

prefer later opening. Estimate how many of the library's

1000

users prefer later opening.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	$600$ users	2	The sample proportion is $30/50=0.6$ . Apply this to the population: $0.6\times1000=600$ .

Tier 2 · Standard

A random sample of

80

items from a production run of

800

contains

18

items with a surface mark. Estimate the number in the whole run with a surface mark and state one limitation of the estimate.

(3)

(Total for Question 2 is 3 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	$180$ items. The sample may differ from the population by chance, so the estimate need not equal the true number.	3	The sample is one tenth of the production run, so scale $18$ by $10$ to get $180$ . Because only a sample was inspected, sampling variation remains even if the selection was random.

Tier 3 · Hard

A service invites a random sample of

450

customers to answer a survey. Only

270

reply, and

189

of the replies support a change. Use the replies to estimate the number of supporters among all

12000

customers, then explain a serious limitation.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	Estimated supporters $=8400$ . Non-response bias may make the replies unrepresentative because the $180$ non-responders may have different views.	4	Among replies, the support proportion is $189/270=0.7$ , giving $0.7\times12000=8400$ . However, the estimate assumes responders and non-responders have similar opinions; the low response rate may break that assumption.

Tier 1 · Easy

A council wants the views of all

6000

residents about a cycle lane and surveys

80

visitors to a sports centre. Write down the population and give one reason why the sample may be unrepresentative.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	The population is all $6000$ residents. Visitors to the sports centre may have different views from residents who do not use it (accept any valid reason that the sports-centre sample may be biased).	2	The group whose views the council wants is all $6000$ residents, so that is the population. The sample comes from only one location and may over-represent people who use the sports centre, so it may not represent all residents.

Tier 2 · Standard

Two random samples estimate support for a new crossing. In a sample of

25

14

people support it. In a sample of

200

104

people support it. Use the more reliable sample to estimate how many of

3000

residents support the crossing. Give a reason for your choice of sample.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$1560$ residents. The sample of $200$ is more reliable because a larger random sample is likely to have less sampling variation.	3	Use the larger random sample. Its support proportion is $\dfrac{104}{200}=0.52$ , so the estimate is $0.52\times3000=1560$ . A larger random sample is generally less affected by chance variation than a sample of $25$ .

Tier 3 · Hard

In a representative sample of

120

households,

78

own at least one pet. In a separate representative sample of

90

pet-owning households,

54

own a dog. Use both samples to estimate the percentage of all households that own a dog. Explain why using

54/90

as the estimate would be wrong.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	Estimated percentage $=39\%$ . $54/90$ estimates the proportion of pet-owning households that own a dog, not the proportion of all households that own a dog.	4	Estimate the proportion of households that own a pet as $78/120=0.65$ . Estimate the proportion of pet-owning households that own a dog as $54/90=0.6$ . Therefore the estimated proportion of all households that own a dog is $0.65\times0.6=0.39$ , or $39\%$ . The second sample alone has pet-owning households as its population, so its $60\%$ cannot be applied to all households.

Tier 2 · Standard

A recycling team uses a sample of

160

households to estimate that

1050

of the district's

3000

households use a food-waste collection. Work out how many households in the sample used the collection. Give one reason why the estimate of

1050

may not equal the true district total.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$56$ households. Households that use the food-waste collection may be more or less likely to appear in the sample — for example, uptake may vary between streets or seasons — so the sample proportion may differ from the district's (accept any valid sampling limitation).	3	The estimated population proportion is $1050/3000=0.35$ . Apply this proportion to the sample: $0.35\times160=56$ . Even a representative sample gives an estimate, because another sample could contain a different proportion.

Tier 3 · Hard

A club emails

320

randomly selected members. Of the

224

who reply,

98

intend to renew their membership. Without making an assumption about the members who did not reply, work out the least and greatest possible percentages of the

320

sampled members who intend to renew. Explain why using

98/224

to describe all club members may be unreliable.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	Least possible percentage $=30.625\%$ (or $30.6\%$ to 1 d.p.); greatest possible percentage $=60.625\%$ (or $60.6\%$ to 1 d.p.). The members who replied may have different intentions from those who did not reply, so the replies may be unrepresentative of all club members.	4	There are $320-224=96$ non-responders. The least possible number intending to renew is $98$ , giving $98/320\times100=30.625\%$ . The greatest is $98+96=194$ , giving $194/320\times100=60.625\%$ . This wide range shows the possible effect of non-response bias.

Tier 3 · Hard

A coach operator surveys

150

passengers who booked their journeys using its app. Of these passengers,

96

say the service was satisfactory. Use the sample to estimate how many of the operator's

3200

weekly passengers are satisfied. Explain why this sampling method could make the estimate unreliable.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	Estimated number satisfied $=2048$ . Passengers who book at a ticket counter or by telephone are excluded, and their experience or opinion of the service may differ from that of app users.	4	The sample proportion satisfied is $96/150=0.64$ , so the estimate is $0.64\times3200=2048$ . The sampling frame contains only app users and excludes passengers using other booking methods, so it may not represent all weekly passengers.

Q10

Tier 3 · Hard

A leisure centre wants the views of all its adult members. Method A uses a random-number generator to select

120

members from the complete membership list and arranges a time for each selected member to answer. Method B telephones members between

10

am and

2

pm on a weekday until

120

members have answered. State which method is more likely to give a representative sample. Explain why, referring to a group Method B is likely to exclude.

(3)

(Total for Question 10 is 3 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	Method A is more likely to give a representative sample. Method B is likely to exclude members who are at work, at college or otherwise unavailable during weekday daytime; their views may differ from those of members who are available then.	3	Choose Method A first. Selecting at random from the complete membership list gives every adult member a chance of selection. Method B favours members who are available during weekday daytime and can omit working members, students and others who are away then, so it is more likely to be biased.

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2019-11	2F	Q24	3	Allowed	Foundation	QP MS
2022-11	2H	Q15	4	Allowed	Higher	QP MS
2019-11	2H	Q4	3	Allowed	Higher	QP MS
2024-11	3H	Q18	4	Allowed	Higher	QP MS
2019-06	2H	Q3	3	Allowed	Higher	QP MS
2022-11	1H	Q5	2	Non-calculator	Higher	QP MS
2022-06	2H	Q14	3	Allowed	Higher	QP MS
2019-06	2F	Q22	3	Allowed	Foundation	QP MS
2019-11	1H	Q11	2	Non-calculator	Higher	QP MS

Other points in S Statistics

Want help turning this into marks?

Bring S1 or any tricky specification point, and we can work through the method and exam wording together.

Book a free intro call