Statistical studies rely on samples to make conclusions about populations. The quality of these conclusions depends on how the sample is selected. The Digital SAT tests your understanding of sampling methods, potential sources of bias, and when conclusions can be generalised.
Core Concepts
Population vs. Sample
- Population: the entire group you want to study.
- Sample: the subset you actually measure.
Random Sampling
A random sample gives every member of the population an equal chance of being selected. This is the gold standard for generalising results.
Types:
- Simple random sample (SRS): every member equally likely.
- Stratified sample: divide population into groups (strata), then randomly sample from each.
- Systematic sample: select every th member from a list.
- Cluster sample: randomly select entire groups (clusters).
Convenience Sampling
Selecting whoever is easiest to reach. Not random → results may be biased.
Sources of Bias
- Selection bias: the sample doesn't represent the population (e.g., surveying only gym members about exercise habits).
- Voluntary response bias: people self-select into the sample (e.g., online polls).
- Nonresponse bias: some selected individuals don't respond.
- Question wording bias: leading questions influence responses.
Generalisation
Results from a random sample can be generalised to the population it was drawn from — but not to a different population.
Strategy Tips
Tip 1: "Random" Is the Key Word
If a sample is randomly selected, conclusions can be generalised. If not, they can't.
Tip 2: Identify the Population
The conclusions apply only to the population the sample was drawn from.
Tip 3: Look for Sources of Bias
Convenience samples, voluntary response, and non-random selection all introduce bias.
Worked Example: SAT-Style
A researcher surveys 500 randomly selected adults in a city about their exercise habits. Can the results be generalised to all adults in the city?
Yes — the sample is random and drawn from the city's adult population.
Worked Example: Example 2
A website posts a poll asking visitors whether they support a new policy. 80% say yes. Is this result likely to represent the general population?
No — this is voluntary response sampling. People with strong opinions are more likely to respond, creating bias.
Worked Example: Example 3
A school surveys all students in one grade. Can results be generalised to the entire school?
No — the sample is only from one grade. Students in other grades may differ.
Practice Problems
Problem 1
A company surveys its employees by randomly selecting 50 from each department. What type of sampling is this?
Problem 2
A radio station asks listeners to call in with their opinions. What type of bias might occur?
Problem 3
A study randomly selects 1000 adults nationwide. Can results be generalised to all adults in the country?
Want to check your answers and get step-by-step solutions?
Common Mistakes
- Thinking any large sample is representative. Size doesn't fix bias — a biased sample of 10,000 is still biased.
- Generalising beyond the population sampled. A study of college students doesn't generalise to all adults.
- Ignoring voluntary response bias. Self-selected samples are almost always biased.
Key Takeaways
Random sampling is essential for generalisation.
Bias = systematic error in how the sample represents the population.
Results can only be generalised to the population sampled.
Large sample size doesn't fix biased sampling methods.
Convenience and voluntary response samples are not random → unreliable for generalisation.