Planning and Data Collection: Random, Systematic, and Stratified Sampling
Welcome to your study guide on sampling methods! Have you ever wondered how pollsters predict election results by asking only a few thousand people, or how quality control testers know an entire batch of crisps tastes great without eating every single bag? The secret lies in sampling. In this chapter, you will learn the three core probability sampling techniques required for your GCSE Statistics exam: Simple Random Sampling, Systematic Sampling, and Stratified Sampling.
Don't worry if this seems a bit technical at first. We will break down every method step-by-step with simple formulas and everyday examples so you can tackle exam questions with total confidence!
Did you know? In 1936, a US magazine surveyed over 2 million people to predict the presidential election but got it completely wrong! Why? Because they only surveyed wealthy people who owned cars and telephones. A representative sample is far more powerful than just a large sample.
1. Key Foundations: Population, Sample, and Sampling Frame
Before jumping into the three methods, let's make sure we know the basic building blocks:
• Population: The entire group of individuals or items that you want to investigate (e.g., all \(800\) students in a school).
• Census: Collecting data from every single member of the population.
• Sample: A smaller group selected from the population to represent the whole.
• Sampling Frame: A complete, numbered list of all members of the population from which the sample is chosen (e.g., the school register from \(1\) to \(800\)).
• Bias: An unfair sample that does not accurately reflect the population.
Key Takeaway: We use a sample instead of a census because it saves time and money, and is often much more practical.
2. Simple Random Sampling
What is it?
In a Simple Random Sample (SRS), every single member of the population has an equal chance of being chosen. It is completely fair and unbiased.
How to do it (Step-by-Step):
1. Obtain a sampling frame and assign a unique number from \(1\) to \(N\) to every member of the population.
2. Use a random number generator (on a calculator, computer, or random number table) to generate numbers.
3. Ignore any duplicate numbers (repeats).
4. Select the individuals whose assigned numbers match the generated random numbers until your sample size \(n\) is reached.
Alternative manual method for small groups: Write all names on identical slips of paper, place them in a hat, shake well, and draw out the required number without looking.
Real-World Analogy
Think of the National Lottery: numbered balls bounce around in a machine, and each ball has the exact same probability of being picked.
Advantages and Disadvantages:
• Advantage: It is completely unbiased; every member has an equal probability of selection.
• Advantage: Free from researcher influence or personal opinion.
• Disadvantage: You need a full sampling frame, which might be hard to get.
• Disadvantage: By pure chance, you could end up with an unrepresentative sample (e.g., accidentally picking only boys from a mixed school).
Key Takeaway: Simple random sampling gives everyone an equal chance. Always remember to state: "ignore repeats" when describing how to use random number tables on the exam!
3. Systematic Sampling
What is it?
Systematic Sampling chooses individuals at regular intervals from an ordered list. Instead of picking every item randomly, you pick items at fixed steps through the population.
How to calculate the Sampling Interval:
To find the regular step size (called the interval, \(k\)), use the formula:
\(k = \frac{\text{Population Size } (N)}{\text{Desired Sample Size } (n)}\)
Step-by-Step Process:
1. Number each member of the population from \(1\) to \(N\).
2. Calculate the interval: \(k = \frac{N}{n}\). (Round down to the nearest whole number if needed).
3. Choose a random starting point between \(1\) and \(k\) (e.g., using a random number generator).
4. Select that first person, and then pick every \(k^{\text{th}}\) person after that until the sample is complete.
Worked Example:
A factory makes \(600\) lightbulbs a day. A quality inspector wants a systematic sample of \(30\) bulbs.
• Step 1: Calculate the interval: \(k = \frac{600}{30} = 20\).
• Step 2: Pick a random start number between \(1\) and \(20\). Let's say we roll a number and get \(7\).
• Step 3: Select bulb \(7\).
• Step 4: Add \(20\) repeatedly: \(7, 27, 47, 67, 87, \dots\) all the way to bulb \(587\). You will have exactly \(30\) bulbs!
Advantages and Disadvantages:
• Advantage: Simple and quick to execute, especially with long lists or queues.
• Advantage: Spreads the sample evenly across the entire population.
• Disadvantage: Can introduce severe bias if the list has a periodic pattern (a repeating cycle) that matches the interval \(k\).
Common Mistake to Avoid: Never start automatically at number \(1\)! In an exam, you must mention choosing a random starting point between \(1\) and \(k\).
Key Takeaway: Systematic sampling selects every \(k^{\text{th}}\) item starting from a randomly chosen position between \(1\) and \(k\).
4. Stratified Sampling
What is it?
In Stratified Sampling, the population is divided into distinct, non-overlapping groups called strata (layers) based on a shared characteristic (such as age, gender, or year group). A random sample is then taken from each stratum in exact proportion to the size of that stratum in the population.
The Stratified Formula:
To calculate how many people to select from a specific group, use the formula:
\(\text{Number in sample from stratum} = \frac{\text{Number in stratum}}{\text{Total population}} \times \text{Total sample size}\)
Step-by-Step Worked Example:
A secondary school has \(500\) students across three Key Stages. The headteacher wants a stratified sample of \(60\) students.
• Key Stage 3: \(250\) students
• Key Stage 4: \(150\) students
• Post-16: \(100\) students
• Total Population (\(N\)): \(250 + 150 + 100 = 500\)
• Desired Sample Size (\(n\)): \(60\)
Calculations:
• Key Stage 3: \(\frac{250}{500} \times 60 = 0.5 \times 60 = 30\) students
• Key Stage 4: \(\frac{150}{500} \times 60 = 0.3 \times 60 = 18\) students
• Post-16: \(\frac{100}{500} \times 60 = 0.2 \times 60 = 12\) students
Check: \(30 + 18 + 12 = 60\). The total matches the required sample size!
Handling Decimals and Rounding:
Often, your calculations will give decimals. Always round to the nearest whole number. If the rounded numbers do not sum to your desired sample size, adjust the stratum whose decimal part was closest to the rounding cut-off.
Advantages and Disadvantages:
• Advantage: It is highly representative; all sub-groups are guaranteed to be represented in the correct proportion.
• Advantage: Prevents minority groups from being missed out by chance.
• Disadvantage: You must know the exact proportions and characteristics of the population beforehand.
• Disadvantage: It is more complex and time-consuming to organize than simple random sampling.
Key Takeaway: Stratified sampling ensures every group is represented proportionally: \(\frac{\text{stratum}}{\text{total}} \times \text{sample size}\). Within each stratum, members are chosen using simple random sampling.
5. Quick Method Comparison Guide
Simple Random Sampling:
• How it works: Every item chosen completely by chance (e.g., random numbers).
• Best used when: The population is uniform, small, or you have a complete digital list.
• Main strength: Totally unbiased and fair.
Systematic Sampling:
• How it works: Choose every \(k^{\text{th}}\) item after a random start.
• Best used when: Items are already in an ordered list or coming down a production line.
• Main strength: Very quick and evenly spread across the list.
Stratified Sampling:
• How it works: Proportional numbers chosen from naturally occurring sub-groups.
• Best used when: The population contains distinct sub-groups that might have different opinions or characteristics.
• Main strength: Guarantees proportional representation for all groups.
6. Summary and Exam Checklist
Before sitting your exam, make sure you can:
• Clearly define population, sample, and sampling frame.
• Explain how to use random number tables/generators to select a simple random sample.
• Calculate the interval \(k = \frac{N}{n}\) for systematic sampling and explain the need for a random starting point.
• Apply the stratified sampling formula accurately and check that your sample numbers add up to the total.
• State at least one advantage and one disadvantage for each sampling method.