Chapter: Sampling (AS 2 Section C: Statistics)
Welcome to the study notes for Sampling! In statistics, we often want to find out information about large groups of people, items, or events. However, checking every single item is usually impossible, expensive, or completely impractical. That is where sampling comes in. Think of it like tasting a single spoonful of soup to check if the entire pot has enough salt — you don't need to drink the whole pot to find out!
Don't worry if you find statistical terminology a bit dry at first. We will break down every single concept into clear, bite-sized pieces with real-world examples.
1. Fundamental Terminology
Before exploring the different ways to choose a sample, let's get our foundational terms straight. Examiners love testing these definitions!
Population vs Sample
• Population: The entire collection of all individuals, items, or units of interest. For example, all \(1,200\) students at a school, or every lightbulb produced in a factory today.
• Census: An investigation where information is collected from every single member of the population.
• Sample: A smaller subset of the population chosen to represent the whole population.
Census vs Sample: The Trade-Off
Why not always conduct a census?
• Advantages of a Census: It is completely accurate and provides a full, unbiased picture because every single member is measured.
• Disadvantages of a Census: It is time-consuming, expensive, can generate an overwhelming amount of data, and cannot be used if the testing process destroys the item (e.g., testing the lifetime of a firework or the crash safety of a car!).
• Advantages of a Sample: It is much quicker, cheaper, easier to manage, and practical for destructive testing.
• Disadvantages of a Sample: Data might not be fully representative, and there is natural variation (sampling error) between different samples.
Sampling Units and Sampling Frames
• Sampling Unit: An individual member or element of the population that can be sampled (e.g., one student, one patient, one manufactured component).
• Sampling Frame: A complete, uniquely identifiable list of all the sampling units in the population (e.g., a school register, an electoral roll, or an employee database with ID numbers).
Memory Tip: The Frame is the picture frame holding the whole list; the Unit is one individual inside that frame!
Key Takeaway: A census studies everyone but costs time and money; a sample studies a selection. To choose a random sample systematically, you almost always need a sampling frame.
2. Random (Probability) Sampling Techniques
In random sampling, every item in the population has a known, non-zero probability of being selected. This eliminates selection bias.
Method A: Simple Random Sampling (SRS)
Definition: A sample of size \(n\) chosen in such a way that every individual in the population, and every possible sample of size \(n\), has an equal chance of being selected.
How to carry it out:
1. Allocate a unique number from \(1\) to \(N\) to each member of the sampling frame.
2. Generate \(n\) distinct random numbers using a random number generator, computer software, or a lottery method (e.g., drawing numbers from a hat).
3. Select the individuals corresponding to those generated numbers. (Ignore repeats!).
Advantages:
• Completely free from personal bias.
• Easy to understand and straightforward for small populations.
• Each selection is independent of the others.
Disadvantages:
• Requires a full sampling frame before you can begin.
• Can be impractical or costly if the population is widely dispersed geographically.
• A purely random draw might accidentally miss smaller sub-groups of interest.
Method B: Systematic Sampling
Definition: A method where items are chosen at regular intervals from an ordered list.
How to carry it out:
1. Number the population from \(1\) to \(N\).
2. Calculate the sampling interval: \(k = \frac{N}{n}\) (round to the nearest whole number if needed).
3. Choose a random starting number \(r\) between \(1\) and \(k\).
4. Select the \(r\)-th item, and then every \(k\)-th item thereafter (i.e., items numbered \(r\), \(r + k\), \(r + 2k\), \(r + 3k\), ...).
Example: If \(N = 500\) and you want a sample of \(n = 50\), then \(k = \frac{500}{50} = 10\). Pick a random number between \(1\) and \(10\). If you pick \(7\), your sample consists of items numbered \(7, 17, 27, 37, \dots, 497\).
Advantages:
• Simple and quick to execute.
• Suitable for large populations.
• Spreads the sample evenly across the entire list.
Disadvantages:
• Requires a sampling frame.
• Common Exam Trap: If the list has a recurring pattern or periodicity that coincides with \(k\) (e.g., every \(7\)-th day is a Sunday, or every \(10\)-th machine item is from an unreliable mold), the sample will be severely biased!
Method C: Stratified Sampling
Definition: The population is divided into mutually exclusive, non-overlapping sub-groups called strata (based on characteristics like age, gender, year group, or income), and a simple random sample is taken from each stratum in proportion to its size in the whole population.
Formula:
The number of items to select from a given stratum is:
\(\text{Number from stratum} = \frac{\text{Size of stratum}}{\text{Total population size } (N)} \times \text{Total sample size } (n)\)
Step-by-Step Worked Example:
A sixth-form college has \(600\) students: \(360\) in Year 13 and \(240\) in Year 14. You want to take a stratified sample of \(50\) students.
• Year 13 sample size: \(\frac{360}{600} \times 50 = 30\) students.
• Year 14 sample size: \(\frac{240}{600} \times 50 = 20\) students.
• Next step: Use simple random sampling within each year group to select the \(30\) and \(20\) specific individuals.
Advantages:
• Guarantees representation from all key sub-groups in exact proportion to their presence in the population.
• More accurate and provides greater precision than simple random sampling when strata differ significantly from one another.
Disadvantages:
• The population must be easily classified into distinct, known strata.
• Requires a complete sampling frame within each individual stratum.
Key Takeaway: Random methods (Simple Random, Systematic, Stratified) eliminate researcher bias, but always require a sampling frame.
3. Non-Random Sampling Techniques
In non-random sampling, items do not have a known or equal chance of selection. These methods are frequently used in market research and opinion polls where obtaining a full sampling frame is impossible or too costly.
Method D: Quota Sampling
Definition: An interviewer or researcher is given a quota of people to interview from specific sub-groups (e.g., interview \(20\) males aged 18–30 and \(30\) females aged 31–50). The interviewer finds people meeting these criteria until each quota is filled.
Stratified vs Quota Sampling (Crucial Difference!):
• Stratified sampling uses random selection within each group and requires a sampling frame.
• Quota sampling leaves the selection within each group up to the researcher and does not require a sampling frame.
Advantages:
• No sampling frame needed.
• Quick, easy, and inexpensive.
• Allows for easy comparison between different groups within the population.
Disadvantages:
• Potential for interviewer bias (e.g., approaching only friendly-looking people).
• Non-respondents are simply replaced by someone else, which can introduce bias.
• Cannot estimate sampling errors statistically because selection is not random.
Method E: Opportunity (Convenience) Sampling
Definition: Taking a sample from people or items that are available at the time and place of the study, and who fit the criteria (e.g., standing outside a shopping center at 10 AM on a Tuesday to ask passers-by questions).
Advantages:
• Very easy, fast, and cheap to carry out.
• Useful for initial pilot studies.
Disadvantages:
• Highly unrepresentative and heavily biased (e.g., surveying people on a Tuesday morning excludes most full-time workers and students).
• Highly dependent on individual researcher convenience.
Key Takeaway: Non-random methods (Quota, Opportunity) do not need a sampling frame, but introduce researcher bias and cannot be analyzed using formal probability theory.
4. Summary Comparison Table
Let's summarize all five sampling methods side by side:
1. Simple Random Sampling
• Type: Random
• Requires Sampling Frame? Yes
• Main Strength: Completely unbiased; mathematically pure.
• Main Limitation: Impractical for large/scattered populations.
2. Systematic Sampling
• Type: Random
• Requires Sampling Frame? Yes
• Main Strength: Very quick and evenly distributed across the list.
• Main Limitation: Severe bias if there is an underlying periodic cycle in the list.
3. Stratified Sampling
• Type: Random
• Requires Sampling Frame? Yes
• Main Strength: Every stratum is proportionally represented.
• Main Limitation: Needs full lists for every single sub-group.
4. Quota Sampling
• Type: Non-Random
• Requires Sampling Frame? No
• Main Strength: Representative proportions without needing a frame.
• Main Limitation: Interviewer selection bias.
5. Opportunity Sampling
• Type: Non-Random
• Requires Sampling Frame? No
• Main Strength: Minimal cost and immediate access to data.
• Main Limitation: Prone to severe bias; rarely representative.
5. Common Pitfalls and Exam Tips
• Remember the random start: When describing systematic sampling, students often forget to state: "Select a random number between \(1\) and \(k\) for the starting position." Missing this loses an easy mark!
• Ignore duplicates: When describing how to use a random number generator for simple random sampling, always state: "Generate random numbers, ignoring any repeats, until \(n\) unique members are selected."
• Rounding in Stratified Sampling: Stratum calculations might result in decimals. Always round to the nearest whole number and ensure the sum of all strata equals the total desired sample size \(n\).
• Identify the context: In exam questions asking you to critique a sampling method, look at the specific context. If an interviewer surveys shoppers on a weekday morning, mention that working people or school children are excluded.
Final Thought: Always ask yourself two simple questions when solving sampling questions: "Do I have a list of everyone (sampling frame)?" and "Is the choice based on pure chance (random) or convenience?" If you can answer these, you will master this chapter!