A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Statistical Sampling: Population and Sample — mark scheme explained
The short answer
In statistics, the terms 'population' and 'sample' are fundamental. Understanding these concepts is crucial for making informed inferences about a larger group based on data collected from a smaller subset. Population: The population refers to the entire group of individuals or items that you are interested in studying.
The question
A researcher wants to estimate the average height of students in a school. The school has 1,000 students. Describe how you would use simple random sampling to select a sample of 50 students.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions on statistical sampling, marks are typically awarded for correct definitions, appropriate use of sampling techniques, and critical evaluation of the methods used. Ensure your answers are clear, concise, and well-structured.
What the command words demand
- Explain
- Provide a detailed account of why or how something happens, including reasons and examples.
- Describe
- Give a detailed account of the characteristics or features of something.
- Discuss
- Consider different viewpoints or aspects of an issue, providing arguments for and against.
- Identify
- Recognize and name specific elements or factors.
- Compare
- Highlight similarities and differences between two or more items or concepts.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate approximately 5 minutes per mark to ensure you have enough time to provide detailed and accurate responses.
- 1. Assign a unique number to each student from 1 to 1,000.1 mark
- 2. Use a random number generator or a table of random numbers to select 50 unique numbers between 1 and 1,000.1 mark
- 3. Identify the students corresponding to these 50 numbers and include them in your sample.1 mark
Final answer: The researcher would assign each student a unique number from 1 to 1,000, use a random number generator to select 50 unique numbers, and then identify the students corresponding to these numbers for the sample.
Work through every step correctly and you earn all 3 marks.
Another worked example
Explain why opportunity sampling might be biased in a study of dietary habits in a diverse city. Provide an example.
- 1. Opportunity sampling involves selecting individuals who are readily available or easy to contact.0 marks
- 2. This method can introduce bias because the sample may not accurately represent the population.2 marks
- 3. For example, if you conduct a survey at a single location like a shopping mall, you might only capture the dietary habits of people who frequent that mall, which may not reflect the broader city's diverse eating patterns.2 marks
Final answer: Opportunity sampling can be biased because it involves selecting individuals who are readily available or easy to contact. For example, conducting a survey at a single shopping mall might only capture the dietary habits of people who frequent that mall, which may not represent the broader city's diverse eating patterns.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Confusing population and sample
Why it happens: Students sometimes mix up the definitions of population and sample, leading to incorrect inferences.
Fix: Always clearly define the population as the entire group of interest and the sample as a subset selected for analysis.
Assuming all samples are representative
Why it happens: Students may assume that any sample will accurately represent the population, ignoring potential biases.
Fix: Understand that different sampling techniques can lead to different conclusions and consider the context when selecting a sample.
Using convenience sampling without acknowledging bias
Why it happens: Students might use opportunity sampling without recognizing its limitations, leading to biased results.
Fix: Always acknowledge and discuss the potential biases introduced by opportunity sampling in your analysis.
Failing to consider sample size
Why it happens: Students may not realize that a larger sample size generally provides more reliable results.
Fix: Understand the importance of sample size in reducing sampling variability and improving the reliability of your inferences.
Not using random number generators for SRS
Why it happens: Students might manually select a sample, thinking it is random, but this can introduce bias.
Fix: Use a random number generator or a table of random numbers to ensure that every member of the population has an equal chance of being selected.
Ignoring sampling variability
Why it happens: Students may not understand that different samples can lead to different conclusions due to natural variability.
Fix: Recognize the importance of understanding and accounting for sampling variability in your analysis.
Failing to critique sampling techniques
Why it happens: Students might not critically evaluate the appropriateness of different sampling techniques for their study.
Fix: Always consider the context and potential biases when selecting a sampling technique and provide a rationale for your choice.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Simple Random Sampling | Describe how to select a random sample of 50 students from 1,000. | 3 |
| Evaluate Sampling Bias | Explain why opportunity sampling causes bias and give a relevant supporting example. | 4 |
| Sampling Probability | Calculate the probability of a student being selected in a simple random sample. | 2 |
| Evaluate Sampling Reliability | Assess how sampling method and sample size affect the reliability of a population inference. | 4 |
| Total across these question types | 13 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.