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A population is the entire group you want to draw conclusions about (e.g. all students in a school).
A sample is a subset of the population that is actually surveyed or measured.
Sampling is used because testing the whole population is often too expensive, time-consuming, or impractical (e.g. testing every light bulb would destroy them).
The sample should be representative of the population so that conclusions drawn from it can be generalised back to the whole population.
Why Sampling is Used
It is cheaper and faster than a census (a survey of the whole population).
It may be impossible to reach every member of the population (e.g. all fish in the ocean).
Testing can be destructive (e.g. checking matches or fireworks).
A well-chosen sample can give accurate enough results without the cost of a full census.
Representative Samples
A representative sample reflects the key characteristics of the population (e.g. same proportion of boys/girls, ages, or locations).
A simple random sample gives each member an equal selection chance. It helps avoid deliberate selection bias but does not guarantee a perfectly representative realised sample.
Use a complete sampling frame and account for nonresponse. Equal selection chances alone cannot remove every source of survey bias.
A larger sample is generally more reliable than a smaller one, but only if it is still representative.
Bias in Sampling
Bias occurs when the sample is not representative, leading to conclusions that are not true for the whole population.
Bias can arise from the sampling method (e.g. only asking people in a shopping centre, which excludes those who don't go there).
Bias can arise from the location of the survey (e.g. asking people outside a gym about exercise habits).
Bias can arise from the time of the survey (e.g. asking commuters at 8am misses night-shift workers).
A convenience sample (e.g. asking friends or people nearby) is often biased because it does not represent the whole population.
A larger sample does NOT remove bias – if the sampling method is biased, increasing the sample size still gives biased results.
Sample Size and Reliability
A larger sample tends to give more reliable estimates because it reduces the effect of random variation.
A small sample may not reflect the population well, so conclusions drawn from it are less reliable.
However, a large sample does not guarantee accuracy if the sample is biased (e.g. only asking one age group).
When commenting on reliability, mention both the sample size and the sampling method.
Estimating Population Totals
To estimate a population total from a sample, scale up the sample result proportionally.
For example, if 30 out of 200 sampled students own a bike, and the school has 1000 students, estimate: \frac{30}{200} \times 1000 = 150 bike owners.
This scaling assumes the sample is representative of the whole population.
Always state the assumption: the sample is representative and unbiased.
Common Misconceptions
Misconception: A larger sample always removes bias. Correction: A biased method stays biased regardless of sample size.
Misconception: A convenience sample is random. Correction: Convenience samples are not random and are often biased.
Misconception: You can generalise from a sample taken from one narrow group. Correction: The sample must represent the whole population.
Misconception: Confusing the sample with the population in calculations. Correction: Always know which group your data comes from.
Misconception: Criticising a survey without explaining how it introduces bias. Correction: Always link the criticism to a specific source of bias.
Typical Exam Questions
Explain why a sample may be biased – identify the source (method, location, time) and suggest an improvement (e.g. use a random sample from the whole population).
Use sample data to estimate a population total – scale up proportionally and state the assumption.
Comment on reliability – discuss sample size and representativeness.
For a simple random sample, assign unique IDs and generate the required number of distinct random IDs. For systematic sampling, use a random starting position then a fixed interval, checking for periodic patterns.
Identify a leading question – a question that suggests a particular answer (e.g. "Don't you agree that school lunches are terrible?") is biased.
Proportional stratified allocation
For a population of 500 with year-group sizes 200, 150 and 150, a sample of 20 allocates 8, 6 and 6.
Select the allocated number at random within each year group. Round coherently when needed so the final sample size is correct.
A proportional stratified sample
Diapos
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1.What is the difference between a population and a sample?
Easy
AA population is the entire set of items of interest, and a sample is a subset of that population.
BA population is a subset of a sample, and a sample is the entire set of items of interest.
CA population is a larger sample, and a sample is a smaller population.
DThere is no difference; they are the same thing.
2.Why do we use samples instead of the whole population?
Easy
ATo save time and money while still getting information about the population.
BTo make results more accurate than using the whole population.
CBecause samples always give the same results as the population.
DTo avoid having to define the population.
3.A school has 1200 students. A sample of 60 students is taken to estimate the mean number of hours spent on homework. What is the population?
Easy
AThe 60 students sampled
BAll 1200 students
CThe mean number of hours
DHomework hours
4.A larger sample always removes bias from a survey.
Easy
True or false?
5.A survey about school meals is conducted by asking students in the canteen at lunchtime. Which of the following is the most likely source of bias?
Medium
AThe sample is only taken at lunchtime, so it may miss students who do not eat school meals.
BThe sample is too large.
CThe sample is too small.
DThere is no bias in this method.
6.Which of the following are ways to reduce bias in a survey? (select all that apply)
Medium
AUse a random sampling method
BEnsure the sample is representative of the population
CUse a larger sample, even if the method is biased
DAvoid leading questions
EOnly survey people who are easy to reach
7.Match each term with its definition.
Medium
Population
Sample
Bias
A part of the population selected for study
The entire group of interest
A systematic error that makes results unrepresentative
8.Arrange the following steps for selecting a random sample of 20 students from a school of 500 in the correct order.
Medium
Assign each student a number from 1 to 500
Use a random number generator to pick 20 distinct numbers
Select the students with those numbers
Plus de questions
9.Match each sampling method to its description.
Easy
Random sample
Systematic sample
Stratified sample
Every member of the population has an equal chance of being chosen
Members are chosen at regular intervals from an ordered list
The population is divided into groups and a proportional number is taken from each group
10.A sample of 100 students chosen from one school can be used to make a reliable estimate about all students in the country.
Easy
True or false?
11.Which of the following are examples of biased sampling methods? (select all that apply)
Medium
AUsing every tenth entry after a random start in a complete, non-periodic school list
BAsking people outside a gym about their exercise habits
CSelecting names from a hat after writing all student names on equal-sized slips
DInterviewing people at a shopping centre on a weekday morning
EUsing a random number generator to pick 50 students from a school register
12.Put the steps for selecting a stratified sample of 20 students from a school with 200 Year 7, 150 Year 8, and 150 Year 9 students in the correct order.
Medium
Calculate the proportion of each year group in the school
Multiply each proportion by 20 to find how many to choose from each year group
List the students in each year group
Choose the required number of students at random from each year group
13.A student wants to estimate the mean height of all students in her school. She measures the heights of students in her class only. Which of the following is the most likely reason this sample is biased?
Medium
AThe sample is too small.
BThe sample is not random and may not represent all students.
CThe sample is too large.
DThe sample includes only boys.
14.A survey asks: 'Don't you agree that the new sports centre is a great idea?' Which of the following best describes the flaw in this question?
Medium
AIt is too long.
BIt is a leading question that encourages a particular answer.
CIt is too difficult to understand.
DIt has too many possible answers.
15.Which methods are generally suitable for reducing selection bias when estimating employees’ commuting times? Select all that apply.
Medium
AAsking employees who arrive early in the morning
BUse a random start, then every fifth employee in a complete list with no relevant periodic pattern
CPosting a survey on the company noticeboard and asking employees to respond voluntarily
DDividing employees into departments and taking a proportional random sample from each department
EUsing a random number generator to select 50 employees from the full employee list
16.A larger sample size always makes a sample representative of the population.
Easy
True or false?
17.A sample of 50 students is taken from a school of 1000 students. 20 of them say they walk to school. Which calculation gives the best estimate for the number of students in the whole school who walk to school?
Medium
A1000 × (20/50)
B50 × (20/1000)
C1000 × (50/20)
D20 × (50/1000)
18.A sample of 200 students is always more reliable than a sample of 50 students, regardless of how the samples are chosen.
Easy
True or false?
19.Which of the following are likely to introduce bias into a survey? (select all that apply)
Medium
AAsking only students in the library
BUsing a random sample from the school register
CAsking a question that suggests a particular answer
DSurveying a large sample
ECollecting responses only from volunteers at an optional after-school assembly
20.Put the steps for selecting a stratified sample of 30 students from a school with 300 Year 7, 200 Year 8, and 100 Year 9 students in the correct order.
Medium
Calculate the number needed from each year group: 15, 10, and 5
List all students in each year group
Randomly select the required number from each year group
Combine the selected students to form the sample
21.A school has 1200 students. A survey of 80 students finds that the mean number of hours spent on homework per week is 5.2 hours. What is the best estimate for the total number of hours spent on homework by all students in the school per week?
Medium
A6240 hours
B5200 hours
C624 hours
D6500 hours
22.A survey about the amount of time students spend on homework is conducted by asking students as they leave the school library at 4 pm. Which of the following is the most likely source of bias?
Medium
AThe sample is taken at a specific time and place, so it may not represent all students.
BThe sample size is too small to be reliable.
CThe question is leading.
DThe sample is random.
23.A survey of 100 students from one school can be used to make a reliable estimate about all students in the country.
Medium
True or false?
24.In statistics, what is the entire group of individuals or items that we want information about called?
Easy
APopulation
BSample
CCensus
DData set
25.A sample is a subset of a population.
Easy
True or false?
26.Which of the following is a reason why a sample is often used instead of the whole population?
Easy
AIt is cheaper and quicker to collect data from a sample
BA sample always gives the exact true value
CA sample avoids all bias
DA sample is always more accurate than a census
27.A school has 1500 students. A sample of 100 students is taken to estimate the mean time spent on homework. What is the population?
Easy
A1500 students
B100 students
C1400 students
DThe mean time spent on homework
28.Match each sampling method with its description.
Medium
Random sample
Stratified sample
Convenience sample
Every member of the population has an equal chance of being chosen
The population is divided into groups and a proportional number is chosen from each group
Members are chosen because they are easy to reach
29.Put the steps for selecting a random sample of 20 students from a school of 500 in the correct order.
Medium
Assign each student a number from 1 to 500
Use a random number generator to pick 20 different numbers
Select the students corresponding to the chosen numbers
30.Put the following steps for selecting a random sample of 30 students from a school of 600 in the correct order.
Medium
Number all students from 1 to 600
Use a random number generator to pick 30 distinct numbers
Select the students with those numbers
Check the sample is representative
31.Which of the following are advantages of using a sample instead of a census? (select all that apply)
Medium
AIt is cheaper
BIt is faster
CIt is always more accurate
DIt can be used when the population is destroyed by testing
32.A sample that is chosen by asking the first 50 people to leave a school assembly is a random sample.
Easy
True or false?
33.A school has 1500 students. A stratified sample of 100 is to be taken using year groups. There are 300 students in Year 7, 400 in Year 8, 500 in Year 9, and 300 in Year 10. How many Year 9 students should be in the sample? Round the proportional allocation to the nearest whole student.
Medium
A33
B50
C40
D25
34.A larger sample size always makes a sample more representative of the population.
Medium
True or false?
35.Put the steps for selecting a stratified sample of 40 students from a school with 200 Year 7, 300 Year 8, and 500 Year 9 students in the correct order.
Medium
Calculate the proportion of each year group
Multiply each proportion by 40 to find the number from each year
Select a random sample of that size from each year group
Combine the samples
36.A school has 1200 students. A survey of 80 students finds that the mean number of hours spent on homework per week is 5.2 hours. What is the best estimate for the total number of hours spent on homework by all students in one week?
Medium
A6240 hours
B5200 hours
C4800 hours
D960 hours
37.A sample that is chosen by asking every 10th person to leave a shopping centre is a random sample.
Medium
True or false?
38.Put the following steps for selecting a random sample of 20 students from a school of 500 in the correct order.
Medium
Obtain a list of all 500 students
Assign each student a number from 1 to 500
Use a random number generator to select 20 distinct numbers
Select the students with those numbers
39.A school has 1200 students. A sample of 80 students is taken to estimate the mean number of hours spent on homework per week. The sample mean is 5.2 hours. What is the best estimate for the total number of hours spent on homework by all students in the school per week?
Medium
A6240 hours
B416 hours
C624 hours
D5200 hours
40.In a survey, 18 out of 60 students said they cycle to school. What fraction of the sample cycles to school?
Easy
A3/10
B1/3
C3/5
D18/100
41.In a survey, 15 out of 50 students said they walk to school. Estimate how many students walk to school out of a population of 800 students.
Medium
42.What is the term for a sample that accurately reflects the characteristics of the population?
Easy
43.Complete the sentence.
Easy
A sample that is not representative may introduce ____ into the results.
44.A survey of 80 randomly chosen students from a school of 1200 found that 32 of them travel by bus. Estimate the number of students in the whole school who travel by bus.
Medium
45.Give one reason why a sample is often used instead of the whole population in a survey.
Medium
46.Fill in the blanks with the correct terms.
Easy
The whole set of items or people being studied is called the ____, and the subset that is actually surveyed is called the ____.
47.In a survey, 12 out of 40 students said they cycle to school. Estimate how many students cycle to school out of a school of 600 students.
Easy
48.A sample of 50 students from a school of 800 found that the mean number of books read per student per year is 6. Estimate the total number of books read by all students in the school in a year.
Medium
49.A survey of 120 students found that 45% of them have a pet. Estimate how many students in a school of 1500 have a pet.
Medium
50.A survey asks: 'Do you agree that the school day should be longer to improve learning?' Identify the main flaw with this question and suggest one improvement.
Medium
51.A stratified sample of 40 students is to be taken from a school with 600 students in Year 9, 500 in Year 10, and 400 in Year 11. How many Year 10 students should be in the sample? (Give your answer to the nearest whole number.)
Medium
52.In a survey, 18 out of 60 students said they cycle to school. Estimate the number of students who cycle to school in a school of 900 students.
Medium
53.A student wants to estimate the average height of all students in her school. She measures the heights of students in her class only. Explain why this sample may be biased and suggest an improvement.
Medium
54.A survey about the amount of time students spend on homework is conducted by asking students as they leave the school library at 4 pm. Explain why this sample may be biased and suggest one improvement.
Hard
55.A student wants to estimate the average height of all students in her school. She measures the heights of students in her class only. Explain why this sample may be biased and suggest one improvement.
Medium
56.A student wants to estimate the average height of all students in her school. She measures the heights of students in her class only. Explain why this sample may not be representative and suggest one improvement.
Hard
57.In a survey, 15 out of 50 students said they walk to school. Estimate how many students walk to school out of a population of 800 students.
Medium
A264 students
B240 students
C216 students
D288 students
58.A survey of 80 randomly chosen students from a school of 1200 found that 32 of them travel by bus. Estimate the number of students in the whole school who travel by bus.
Medium
A576 students
B528 students
C432 students
D480 students
59.In a survey, 12 out of 40 students said they cycle to school. Estimate how many students cycle to school out of a school of 600 students.
Easy
A198 students
B162 students
C180 students
D216 students
60.A sample of 50 students from a school of 800 found that the mean number of books read per student per year is 6. Estimate the total number of books read by all students in the school in a year.
Medium
A5280 books
B5760 books
C4800 books
D4320 books
61.A survey of 120 students found that 45% of them have a pet. Estimate how many students in a school of 1500 have a pet.
Medium
A811 students
B675 students
C743 students
D607 students
62.A stratified sample of 40 students is to be taken from a school with 600 students in Year 9, 500 in Year 10, and 400 in Year 11. How many Year 10 students should be in the sample? (Give your answer to the nearest whole number.)
Medium
A15 students
B14 students
C12 students
D13 students
63.In a survey, 18 out of 60 students said they cycle to school. Estimate the number of students who cycle to school in a school of 900 students.
Medium
A297 students
B324 students
C243 students
D270 students
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