SPEC 1.1 • STATISTICAL SAMPLING
Sampling Methods and Techniques
Learn how to select representative samples using different sampling methods and evaluate their effectiveness
Why Sample?
In statistics, we often want to learn about a large group of people or items. We have two options:
Census
Collecting data from every single member of the population.
Problems:
Time-consuming
Expensive
May be impractical or impossible
Data may be out of date by completion
Sample
Collecting data from a subset of the population.
Benefits:
Quicker to collect
Less expensive
More practical
Can still give accurate results
Key Concept
A good sample should be representative of the population - it should reflect the characteristics of the whole group.
Key Terminology
Population
The entire group of individuals or items that we want to study or make conclusions about.
Sampling Frame
A list of all members of the population from which a sample can be selected. For example: a school register, electoral roll, or customer database.
Sample
A subset of the population that is selected for study.
Important
Bias occurs when certain members of the population are more likely to be selected than others, leading to a sample that is not representative.
Simple Random Sampling
Definition
Every member of the population has an equal chance of being selected, and each selection is independent of the others.
How to carry out simple random sampling:
1
Obtain a sampling frame (numbered list of all members)
2
Assign each member a unique number
3
Use a random number generator or random number table to select the required number of members
Advantages
Bias-free if done correctly
Easy to understand
Each member has equal chance
Disadvantages
Needs a complete sampling frame
Can be time-consuming for large populations
May not represent subgroups well
01Worked Example
A school has 850 students numbered 001 to 850. A researcher wants to select a simple random sample of 20 students using a random number table. The first line of the table reads: 847291063584127930...
Which students would be selected from this line?
1
Read the digits in groups of 3 (since we need 001-850)
2
847, 291, 063, 584, 127, 930...
3
Accept numbers from 001 to 850, reject numbers outside this range
4
847 ✓, 291 ✓, 063 ✓, 584 ✓, 127 ✓, 930 ✗ (too large)
AnswerStudents 847, 291, 063, 584, and 127 would be selected from this line. Continue reading the table for remaining 15 students.
Systematic Sampling
Definition
Members are selected at regular intervals from a sampling frame, starting from a randomly chosen point.
Formula
where N = population size, n = sample size, k = sampling interval
Method:
1
Calculate the sampling interval: k = N ÷ n
2
Randomly select a starting point between 1 and k
3
Select every k-th member from that point
Exam Tip
If k is not a whole number, round down to ensure you can select enough members. You may need to state your method clearly in the exam.
Worked Example02
A factory has 480 workers. A sample of 30 workers is needed for a health survey. Describe how to select a systematic sample.
1
Calculate the sampling interval: k = N ÷ n = 480 ÷ 30 = 16
2
Number all workers from 1 to 480
3
Randomly select a starting point between 1 and 16 (e.g., 7)
4
Select every 16th worker from the starting point
AnswerIf starting at 7, select workers: 7, 23, 39, 55, 71, 87, ... continuing until 30 workers are selected.
Stratified Sampling
Definition
The population is divided into distinct groups (strata) based on a characteristic, then a random sample is taken from each stratum in proportion to its size.
Formula
When to use: When you have distinct subgroups (e.g., age groups, year groups, gender) and want to ensure each is fairly represented.
Advantages
Guarantees representation of all subgroups
More accurate than simple random sampling
Allows comparison between strata
Disadvantages
Requires knowledge of population structure
Strata must be clearly defined
More complex to organise
Worked Example03
A sixth form has 180 students in Year 12 and 140 students in Year 13. A stratified sample of 40 students is required. How many students should be selected from each year group?
1
Total population = 180 + 140 = 320 students
2
Year 12: (180 ÷ 320) × 40 = 0.5625 × 40 = 22.5
3
Year 13: (140 ÷ 320) × 40 = 0.4375 × 40 = 17.5
4
Round appropriately: Year 12 = 23, Year 13 = 17 (or 22 and 18)
AnswerSelect 23 students from Year 12 and 17 students from Year 13 (totalling 40).
Worked Example04
A company has employees in three departments: Sales (120), Marketing (80), and IT (50). A stratified sample of 50 employees is needed. Calculate the sample size from each department.
1
Total population = 120 + 80 + 50 = 250 employees
2
Sales: (120 ÷ 250) × 50 = 0.48 × 50 = 24
3
Marketing: (80 ÷ 250) × 50 = 0.32 × 50 = 16
4
IT: (50 ÷ 250) × 50 = 0.20 × 50 = 10
5
Check: 24 + 16 + 10 = 50 ✓
AnswerSales: 24, Marketing: 16, IT: 10
Quota Sampling
Definition
The researcher divides the population into groups and then selects a specified number (quota) from each group. Unlike stratified sampling, the selection within each quota is not random.
How it works:
1
Population is divided into categories (e.g., age groups, gender)
2
Quotas are set for each category
3
Researcher selects people until each quota is filled
4
Selection within quotas is at researcher's discretion
Advantages
Quick and cheap
No sampling frame needed
Ensures key groups are represented
Disadvantages
Not random - potential for bias
Researcher may select "convenient" people
Cannot calculate sampling error
Opportunity (Convenience) Sampling
Definition
The researcher selects people who are readily available and willing to participate. This is the easiest but often least reliable method.
Examples:
Surveying people walking past in a shopping centre
Asking students in your own class
Interviewing people at a specific location
Advantages
Very quick and easy
Low cost
Useful for pilot studies
Disadvantages
Highly likely to be biased
Not representative
Results cannot be generalised
Exam Warning
Opportunity sampling is rarely appropriate for serious research due to its high potential for bias.
05Worked Example
Identify the sampling method used in each scenario:
(a) A researcher stands outside a supermarket and asks the first 50 people who agree to answer questions about shopping habits.
(b) A company uses employee ID numbers and a random number generator to select 100 employees for a survey.
(c) A polling company ensures they interview 200 men and 200 women for a political survey.
(a)
Opportunity sampling - researcher selects whoever is available and willing
(b)
Simple random sampling - using random number generator with a numbered list (sampling frame)
(c)
Quota sampling - set quotas for each gender, but selection within quotas is not random
Note: If (c) used random selection within each gender group from a complete list, it would be stratified sampling.
Comparing Sampling Methods
MethodRandom?Needs Sampling Frame?Best For
Simple RandomYesYesHomogeneous populations
SystematicPartiallyYes (ordered list)Large populations with list
StratifiedYesYesPopulations with distinct subgroups
QuotaNoNoQuick surveys, market research
OpportunityNoNoPilot studies only
Exam Tip
In exams, you may be asked to explain why a particular method is suitable or unsuitable for a given situation. Always link your answer to the specific context given.
06Worked Example
A researcher wants to investigate study habits of university students. She decides to survey students leaving the library between 9am and 10am on a Monday morning.
(a) Identify the sampling method used.
(b) Explain one source of bias in this sample.
(a)
Opportunity sampling - selecting students who happen to be available at a specific time and place
(b)
Sources of bias:
Students using the library early Monday may be more dedicated/organised - not representative of all students
Students with lectures at this time will be excluded
Students who study in other locations (home, cafes) are excluded
Part-time students or those with jobs may not be present
Conclusion: The sample is likely to over-represent conscientious, full-time students and under-represent others.
07Worked Example
A gym wants to survey its 2400 members about new facilities. They want a sample of 120 members. The membership is divided into: Standard (1500), Premium (700), and VIP (200).
(a) Suggest an appropriate sampling method and justify your choice.
(b) Calculate how many members should be selected from each category.
(a)
Stratified sampling - because there are distinct membership categories that may have different views on facilities, and we want proportional representation of each group.
(b)
Standard: (1500 ÷ 2400) × 120 = 75
•
Premium: (700 ÷ 2400) × 120 = 35
•
VIP: (200 ÷ 2400) × 120 = 10
•
Check: 75 + 35 + 10 = 120 ✓
AnswerStandard: 75, Premium: 35, VIP: 10
Practice
Question 1: A college has 1200 students. A researcher wants a sample of 60 students using systematic sampling. Calculate the sampling interval and describe the selection process.
Step 1: Sampling interval k = N ÷ n = 1200 ÷ 60 = 20
Step 2: Number all students 1 to 1200
Step 3: Randomly select a starting point between 1 and 20
Step 4: Select every 20th student from that point
Example: If starting point is 8, select students: 8, 28, 48, 68, 88, ... up to 60 students.
Practice
Question 2: A company employs 240 men and 160 women. Calculate the number of each gender that should be included in a stratified sample of 50 employees.
Total population: 240 + 160 = 400
Men: (240 ÷ 400) × 50 = 0.6 × 50 = 30
Women: (160 ÷ 400) × 50 = 0.4 × 50 = 20
Check: 30 + 20 = 50 ✓
Answer30 men and 20 women
Practice
Question 3: Explain one advantage and one disadvantage of using quota sampling compared to stratified sampling.
Advantage of quota sampling:
Does not require a complete sampling frame, making it quicker and cheaper to conduct, especially for large populations where a full list of members is not available.
Disadvantage of quota sampling:
Selection within quotas is not random, so it is subject to researcher bias. The researcher may unconsciously select people who appear approachable or similar to themselves, leading to a non-representative sample.
Summary
Key Sampling Methods
MethodKey Feature
Simple RandomEqual chance for all, uses random numbers
SystematicEvery k-th member, k = N/n
StratifiedProportional samples from each subgroup
QuotaSet quotas, non-random selection
OpportunityWhoever is available - high bias risk
Key Exam Skills
Calculate stratified sample sizes using proportions
Calculate systematic sampling interval (k = N/n)
Identify sampling methods from descriptions
Evaluate advantages, disadvantages, and sources of bias
Next: Why Sample?