Sampling Methods
This lesson covers sampling methods as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to understand the difference between a census and a sample, know the main types of sampling, and be able to discuss advantages and disadvantages of each method.
Census vs Sample
A census is a survey that collects data from every member of the population.
A sample is a survey that collects data from a subset of the population.
| Feature | Census | Sample |
|---|
| Coverage | Every member of the population | A selection of members |
| Accuracy | Gives a completely accurate result (no sampling error) | Subject to sampling error |
| Cost | Very expensive and time-consuming | Cheaper and quicker |
| Practicality | Often impractical for large populations | Practical for most situations |
| Destructive testing | Not possible (e.g. testing light bulbs to failure) | Suitable for destructive testing |
Key Terminology
| Term | Definition |
|---|
| Population | The whole set of items that are of interest |
| Sample | A subset of the population used to collect data |
| Sampling unit | Each individual member of the population that can be sampled |
| Sampling frame | A list of all sampling units (e.g. the electoral register, school roll) |
| Statistic | A quantity calculated from a sample (e.g. sample mean) |
| Parameter | A quantity that describes a characteristic of the whole population (e.g. population mean) |
Exam Tip: A census gives accurate results but is time-consuming, expensive, and sometimes destructive (e.g. testing the lifetime of batteries would destroy every single one). A sample is quicker, cheaper, and the only option when testing is destructive.
The Sampling Frame
A sampling frame is a list (or database) of every member of the population from which the sample is drawn. For example:
- The electoral register is a sampling frame for voters.
- A school register is a sampling frame for students in a school.
- A telephone directory is a sampling frame for households with listed phone numbers.
For a sample to be representative, the sampling frame must be accurate and up to date. If some members of the population are missing from the frame, the sample will be biased.
Simple Random Sampling
In simple random sampling, every member of the population has an equal chance of being selected. Every possible sample of a given size is equally likely.
How to carry out simple random sampling
- Obtain a complete sampling frame (a numbered list of all N members of the population).
- Use a random number generator or a table of random numbers to select n distinct numbers between 1 and N.
- The members of the population corresponding to those numbers form your sample.
Advantages
- Bias-free -- every member has an equal chance of selection.
- Simple to understand and implement with a random number generator.
- Results can be analysed using standard statistical techniques.
Disadvantages
- Requires a complete and accurate sampling frame, which may not be available.
- Can be impractical for large populations (listing every member may be difficult).
- May not be representative if the sample size is small -- by chance, certain subgroups may be over- or under-represented.
Systematic Sampling
In systematic sampling, you select members at regular intervals from an ordered list.
How to carry out systematic sampling
- Number all N members of the population from 1 to N.
- Calculate the sampling interval: k = N / n (where n is the desired sample size). Round k to the nearest whole number if necessary.
- Choose a random starting point: pick a random number r between 1 and k.
- Select every k-th member starting from r. So you choose members numbered r, r + k, r + 2k, r + 3k, and so on.
Example
A factory produces 500 items per day. You want a sample of 25 items.
- k = 500 / 25 = 20
- Choose a random start, say r = 7.
- Select items numbered 7, 27, 47, 67, 87, ... , 487.
Advantages
- Simple and quick to use once you have the ordered list.
- Spreads the sample evenly across the population.
- Does not require a full list of random numbers.
Disadvantages
- Requires a sampling frame (an ordered list).
- If there is a hidden pattern in the list with the same period as k, the sample will be biased. For example, if every 20th item comes from the same machine, the sample may only include items from one machine.
- Not truly random -- once the starting point is chosen, the rest is determined.
Stratified Sampling
In stratified sampling, you divide the population into strata (distinct subgroups that do not overlap), then take a random sample from each stratum in proportion to the stratum's size.
The formula for the number to sample from each stratum
Number from stratum = (number in stratum / total population) x total sample size
Example
A school has 600 students: 200 in Year 12 and 400 in Year 13. You want a stratified sample of 60 students.
- Year 12: (200 / 600) x 60 = 20 students
- Year 13: (400 / 600) x 60 = 40 students
Then use simple random sampling within each year group to select the required number.
Advantages
- Guarantees proportional representation of each subgroup.
- Gives a more representative sample than simple random sampling, especially when strata differ from each other.
- Reduces sampling variability (the estimate is more precise).
Disadvantages
- You need to know the composition of the population (how many are in each stratum).
- Requires a sampling frame for each stratum.
- More complex to organise than simple random sampling.
- If strata are not clearly defined, there can be ambiguity about which stratum a member belongs to.
Quota Sampling
In quota sampling, the interviewer is given a quota -- a specified number of people to survey from each subgroup. The interviewer then selects people who fit the required profile until the quota is filled.
How it works
- Decide the subgroups and how many people to survey from each.
- The interviewer approaches potential respondents and checks whether they meet the criteria.
- Once the quota for a subgroup is met, no more people from that subgroup are surveyed.
Example
A market researcher needs to survey 100 people: 50 males and 50 females. The researcher stands in a shopping centre and asks people until they have 50 of each.
Advantages
- Does not require a sampling frame -- useful when one is not available.
- Quick and cheap to carry out.
- Guarantees the correct proportions of each subgroup are represented.
Disadvantages
- Not random -- the interviewer chooses who to approach, which introduces bias.
- People who are more visible, more approachable, or more willing to participate are more likely to be selected (selection bias).
- Results may not be generalisable to the whole population because of the non-random selection.
Opportunity (Convenience) Sampling
In opportunity sampling (also called convenience sampling), you simply select people who are available at the time of the study. You use whoever happens to be around.
How it works
- The researcher approaches the first people they encounter who are willing to participate.
- There is no systematic method of selection.
Example
A student stands outside the school canteen at lunchtime and asks the first 30 people who walk past to complete a questionnaire.
Advantages
- Very easy and cheap to carry out.
- Useful for pilot studies or when time and resources are extremely limited.
- Does not require a sampling frame.
Disadvantages
- Highly unlikely to be representative -- you only sample people who happen to be in a particular place at a particular time.
- Strong risk of bias -- certain types of people may be over- or under-represented.
- Results cannot be generalised to the wider population.
Comparing Sampling Methods -- Summary Table
| Method | Random? | Needs sampling frame? | Proportional? | Main advantage | Main disadvantage |
|---|
| Simple random | Yes | Yes | Not guaranteed | Bias-free | Needs complete list |
| Systematic | Partially (random start) | Yes | Not guaranteed | Even spread | Patterns can cause bias |
| Stratified | Yes (within strata) | Yes (for each stratum) | Yes | Proportional representation | Complex to set up |
| Quota | No | No | Yes (by design) | No sampling frame needed | Selection bias |
| Opportunity | No | No | No | Quick and easy | Very biased |
Sampling and Data Types
When designing a sampling method, you should consider the type of data being collected:
- Quantitative data -- numerical values (e.g. height, weight, test score). Can be discrete (countable, e.g. number of siblings) or continuous (measurable, e.g. time in seconds).
- Qualitative data -- non-numerical, descriptive data (e.g. eye colour, favourite subject).
- Primary data -- data you collect yourself for your specific purpose.
- Secondary data -- data that has already been collected by someone else (e.g. government statistics, published research).
Exam Tip: The exam may ask you to recommend a sampling method for a given scenario. Consider whether a sampling frame is available, whether the population can be divided into clear strata, whether the situation requires random selection, and any practical constraints (cost, time, access).
Common Exam Questions on Sampling
Type 1: "Explain why a census might not be appropriate."
Answer: A census surveys every member of the population. This is expensive, time-consuming, and impractical for large populations. If the testing process is destructive, a census would destroy the entire population.
Type 2: "Describe how to take a stratified sample of 50 from a population of 1000 with 600 in group A and 400 in group B."
Answer: Divide the population into the two strata. Group A: (600/1000) x 50 = 30. Group B: (400/1000) x 50 = 20. Use simple random sampling within each group to select 30 from A and 20 from B.
Type 3: "Give one advantage and one disadvantage of opportunity sampling."
Answer: Advantage -- it is quick and easy to carry out. Disadvantage -- it is likely to be biased and not representative of the population.
Summary
- A census surveys every member of the population; a sample surveys a subset.
- Simple random sampling gives every member an equal chance of selection and requires a complete sampling frame.
- Systematic sampling selects at regular intervals from an ordered list (calculate k = N/n and pick a random start).
- Stratified sampling divides the population into strata and samples proportionally from each -- guaranteeing representation.
- Quota sampling requires the interviewer to fill quotas from each subgroup -- no sampling frame needed but not random.
- Opportunity sampling selects whoever is available -- quick but highly biased.
- In the exam, always justify your choice of sampling method in the context of the question.