Random and Systematic Sampling –

Random and systematic sampling are tested on GCSE Maths papers at Foundation and Higher tier across AQA, Edexcel and OCR. Understanding different sampling methods is essential because the way data is collected affects how reliable the conclusions are. Exam questions ask you to identify sampling methods, describe how to carry them out, and explain their advantages and disadvantages. This guide covers the main methods clearly with real-world examples and examiner expectations. For a full overview of every topic, see our complete GCSE Maths topics list.

What Is Sampling?

Sampling means selecting a smaller group (a sample) from a larger group (the population) to collect data from. Since it is usually impractical to survey an entire population, a well-chosen sample allows you to draw conclusions about the whole group.

Random Sampling

In simple random sampling, every member of the population has an equal chance of being selected. Methods include drawing names from a hat, using a random number generator, or assigning numbers and selecting randomly.

Systematic Sampling

In systematic sampling, you select every nth item from an ordered list, starting from a randomly chosen position.

For example, to select 20 from a list of 200: interval = 200 ÷ 20 = 10. Pick a random start between 1 and 10, then select every 10th person.

Stratified Sampling

In stratified sampling, the population is divided into groups (strata) and a proportional number is randomly selected from each group.

Step-by-Step Method

Carrying Out Random Sampling

1. List or number every member of the population.
2. Use a random number generator (or names in a hat) to select the required number.
3. Ensure each member has an equal chance of selection.

Carrying Out Systematic Sampling

1. Calculate the sampling interval: population ÷ sample size.
2. Choose a random starting point between 1 and the interval.
3. Select every nth person from that starting point.

Carrying Out Stratified Sampling

1. Divide the population into relevant groups (e.g. by age, gender, year group).
2. Calculate how many to sample from each group using the formula.
3. Randomly select that number from each group.

Worked Example 1 — Foundation Level

Question: A school has 800 students and wants to survey 40 about school lunches. Describe how to take a simple random sample.

Working:

1. Assign each student a number from 1 to 800.
2. Use a random number generator to produce 40 different numbers between 1 and 800.
3. Survey the students whose numbers are selected.

Answer: Number all 800 students, then use a random number generator to select 40 different numbers. Survey those 40 students. Every student has an equal chance of being chosen.

Worked Example 2 — Higher Level

Question: A factory produces 3000 items per day. Quality control wants to test 100. Describe how to take a systematic sample and state one advantage and one disadvantage.

Working:

Sampling interval = 3000 ÷ 100 = 30. Choose a random number between 1 and 30 (e.g. 14). Test items number 14, 44, 74, 104, ... and so on every 30th item.

Advantage: Easy to carry out on a production line — no need for a full list of items.

Disadvantage: If there is a regular pattern in production (e.g. every 30th item comes from the same machine), the sample could be biased.

Answer: Test every 30th item starting from a random position. Advantage: simple to implement. Disadvantage: may produce a biased sample if the production process has a periodic pattern.

Worked Example 3 — Exam Style

Question: A college has 600 students: 200 in Year 12 and 400 in Year 13. A stratified sample of 60 is required. (a) How many from each year group? (b) Explain why stratified sampling is better than random sampling here.

Working:

(a) Year 12: (200 ÷ 600) × 60 = 20 students. Year 13: (400 ÷ 600) × 60 = 40 students.

(b) Stratified sampling ensures both year groups are represented in proportion to their size. Random sampling might, by chance, select too many from one year and too few from the other, giving an unrepresentative sample.

Answer: (a) 20 from Year 12, 40 from Year 13. (b) Stratified sampling guarantees proportional representation, reducing the risk of an unbalanced sample.

Common Mistakes

• Confusing random with haphazard. "Random" in maths means every member has an equal chance — it does not mean picking whoever is convenient.
• Not calculating the interval for systematic sampling. Always divide population by sample size and show this working.
• Rounding errors in stratified sampling. The numbers from each stratum must be whole numbers and should add up to the total sample size. Round sensibly and adjust if needed.

Exam Tips

• Know the names and definitions of all three methods — random, systematic and stratified.
• Be ready to describe the process step by step — examiners want method, not just the name.
• When asked for advantages and disadvantages, give context-specific answers where possible.
• "Convenience sampling" (e.g. asking friends) is biased — state this if asked to criticise a method.
• For more on sampling, see sampling methods. For data presentation, see bar charts, pie charts and pictograms.

Practice Questions

Q1 (Foundation): Explain why asking only Year 11 students about school uniform is not a random sample of all students' views.

Q2 (Foundation): A dentist has 500 patients and wants to survey 50 using systematic sampling. What is the sampling interval?

Q3 (Higher): A company has 120 office staff, 60 warehouse staff and 20 managers. A stratified sample of 40 is needed. How many from each group?

Practise random and systematic sampling for free on GCSEMathsAI.

Related Topics

• Sampling Methods
• Misleading Graphs and Data
• Probability Basics and Relative Frequency

Summary

Sampling means selecting a smaller group to represent a larger population. In random sampling, every member has an equal chance of selection. In systematic sampling, you select every nth item from a list using the formula interval = population ÷ sample size. In stratified sampling, you divide the population into groups and select proportionally from each. Each method has advantages and disadvantages — random is unbiased but can be impractical; systematic is easy to implement but risks periodic bias; stratified guarantees representation but requires knowledge of the population structure. Always describe the process step by step in exam answers.