Unit AS 2: Applied Mathematics (Section B: Statistics) — Statistical Sampling
Welcome to your study notes for Statistical Sampling. In AS 2 Applied Mathematics, statistics helps us understand the world using data. But before we can calculate averages, probabilities, or test hypotheses, we need to collect that data. How we choose our data matters enormously — get it right, and your conclusions are reliable; get it wrong, and your results can be completely misleading!
Don't worry if you find statistics wordy or tricky at first. This guide breaks down every definition, method, calculation, and exam trap step by step.
1. Core Foundations: Populations, Samples, and Frames
Before looking at sampling methods, let's understand the basic vocabulary that examiners test regularly.
Population vs. Sample
Population: The entire collection of individuals, items, or observations about which information is sought.
Analogy: Imagine the population is an entire swimming pool of soup.
Census: An investigation or survey that collects data from every single member of the target population.
• Advantages: Completely accurate; no sampling bias or sampling error; truly represents the whole population.
• Disadvantages: Extremely time-consuming; very expensive; impractical for large or infinite populations; impossible if testing is destructive (e.g., testing the lifespan of lightbulbs or crash-testing cars — if you test them all, you destroy your entire product!).
Sample: A subset or smaller selection of individuals or items taken from the population, used to make inferences about the whole population.
Analogy: Taking a single spoonful of soup to taste if the whole pot needs salt.
• Advantages: Quicker and cheaper than a census; manageable amount of data; practical when tests are destructive.
• Disadvantages: May introduce sampling bias; natural variation/sampling error (does not give information about every single individual).
Units and Frames
Sampling Units: The individual elements of a population that can be sampled (e.g., a single student, a household, a manufactured battery).
Sampling Frame: A complete, indexed list or register containing all individual sampling units within the population from which a sample is drawn (e.g., a school register, a factory batch log, an electoral register).
Key Takeaway: If you want to know about everything, that is a census. If you take a part of it, that is a sample. To pick items mathematically, you usually need a numbered list called a sampling frame.
2. Random (Probability) Sampling Methods
In random sampling, every item in the population has a known, non-zero chance of being selected. This eliminates personal bias from the researcher.
Method 1: Simple Random Sampling (SRS)
Definition: A sampling method where every member of the population has an equal chance of being selected, and every possible sample of size \(n\) has an equal chance of being chosen.
How to carry out Simple Random Sampling (Step-by-Step):
1. Allocate a unique number from \(1\) to \(N\) to every sampling unit in the sampling frame.
2. Generate random numbers using a random number table, calculator, or computer software.
3. Crucial Exam Step: Ignore any duplicate numbers and numbers outside the range \(1\) to \(N\).
4. Select the corresponding units until your required sample size \(n\) is reached.
Pros & Cons:
• Advantages: Completely unbiased and objective; each unit has an equal chance of selection.
• Disadvantages: Requires a complete sampling frame; can be costly and impractical if the population is geographically spread out.
Method 2: Systematic Sampling
Definition: A method where elements are chosen at regular, fixed intervals from an ordered list.
How to carry out Systematic Sampling (Step-by-Step):
1. Number the units in the sampling frame from \(1\) to \(N\).
2. Calculate the sampling interval \(k = \frac{N}{n}\), where \(N\) is the population size and \(n\) is the desired sample size.
3. Select a random starting point \(r\) from \(1\) to \(k\).
4. Select every \(k^{\text{th}}\) item thereafter: the units chosen will be \(r, r+k, r+2k, r+3k, \dots\) until you have \(n\) units.
Pros & Cons:
• Advantages: Simple and quick to execute; spreads the sample evenly across the entire list.
• Disadvantages: Requires a sampling frame; introduces severe bias if the list has a periodic or cyclic pattern that matches the interval \(k\) (e.g., picking every \(7^{\text{th}}\) day of sales records might always select a Sunday).
Method 3: Stratified Sampling
Definition: The population is divided into mutually exclusive and exhaustive sub-groups called strata (e.g., by age, year group, gender). A simple random sample is then taken independently from each stratum in proportion to its size in the whole population.
The Stratified Allocation Formula:
For any stratum \(i\):
\(\text{Number to select from stratum } i = \frac{\text{Size of stratum } i}{\text{Total population } N} \times \text{Sample size } n\)
Pros & Cons:
• Advantages: Guarantees representation of all sub-groups in exact proportion to the population; reduces sampling error across diverse groups.
• Disadvantages: Requires an accurate sampling frame with details of each unit's sub-group; complex to organise if strata are difficult to define.
Stratified Sampling: Step-by-Step Worked Example
A Sixth Form has \(600\) students in total: \(350\) in Year 13 and \(250\) in Year 14. A stratified sample of size \(n = 48\) is required.
Step 1: Calculate the proportion for Year 13:
\(\text{Year 13 sample} = \frac{350}{600} \times 48 = 28\)
Step 2: Calculate the proportion for Year 14:
\(\text{Year 14 sample} = \frac{250}{600} \times 48 = 20\)
Step 3: Check total: \(28 + 20 = 48\). Exactly \(28\) students are chosen randomly from Year 13, and \(20\) randomly from Year 14.
Note on rounding: If your calculations produce decimal values, round to whole numbers while ensuring the sum of all strata equals the total sample size \(n\).
3. Non-Random (Non-Probability) Sampling Methods
In non-random sampling, the probability of selecting any specific individual is unknown, and the researcher selects the participants.
Method 4: Quota Sampling
Definition: The population is divided into groups according to specific characteristics (similar to strata). The researcher is given a quota of individuals to interview/select from each group until each quota is filled.
Pros & Cons:
• Advantages: No sampling frame needed; quick, cheap, and easy to conduct; allows easy comparison between groups.
• Disadvantages: Non-random selection introduces researcher bias (e.g., interviewer may choose friendly-looking people); non-responses are simply ignored rather than recorded.
Method 5: Opportunity / Convenience Sampling
Definition: Taking samples from members of the population who are readily available at the time of the study and fit the criteria (e.g., surveying the first \(30\) people you meet outside a supermarket on a Monday morning).
Pros & Cons:
• Advantages: Extremely easy, cheap, and convenient to carry out.
• Disadvantages: Highly unrepresentative; contains substantial selection bias based on time and location.
4. Common Exam Pitfalls & Examiner Tips
Make sure you learn these common exam tripwires to secure full marks in AS 2:
1. Confusing Stratified Sampling with Quota Sampling:
Both divide the population into categories by characteristics. However:
• Stratified Sampling is random: It requires a sampling frame and selects units using random numbers within each stratum.
• Quota Sampling is non-random: It does not require a sampling frame; the interviewer chooses who to include until the quota is filled.
2. Forgetting to Mention "Ignoring Duplicates" in Simple Random Sampling:
When asked to describe how to generate a simple random sample using a random number generator, you must state that repeated/duplicate numbers and numbers outside the range are ignored/discarded.
3. Starting Point in Systematic Sampling Must Be Random:
Never say "start with item \(1\)". The starting point \(r\) must be selected at random between \(1\) and \(k\).
4. Contextualise Advantages and Disadvantages:
Examiners often penalise generic answers. If a question asks why a sample is used rather than a census when testing lightbulb lifespans, write: "Testing a lightbulb destroys it (destructive testing), so a census would leave no lightbulbs to sell."
5. Quick Summary Table
Simple Random: Random | Requires Sampling Frame? Yes | Key Feature: Every item & sample has an equal chance.
Systematic: Random | Requires Sampling Frame? Yes | Key Feature: Regular intervals \(k = \frac{N}{n}\) starting from random \(r\).
Stratified: Random | Requires Sampling Frame? Yes | Key Feature: Proportional representation from distinct strata.
Quota: Non-Random | Requires Sampling Frame? No | Key Feature: Researcher fills quotas for specific groups.
Opportunity: Non-Random | Requires Sampling Frame? No | Key Feature: Readily available people/items sampled on the spot.