Random samples, bias, margin of error, and what conclusions are valid.
Concept
To draw conclusions about a population from a sample, the sampling method matters more than the sample size:
Random sample — every member of the population has an equal chance of being chosen. Required for unbiased generalisation.
Biased sample — systematically over- or under-represents some group. Common causes: convenience sampling, voluntary response, leading questions.
Larger random samples shrink the margin of error (roughly proportional to 1/√n).
Margin of error gives a plausible range for the true population parameter. A poll reading "55% ± 3%" means the true value is most likely between 52% and 58%.
Two cautions about conclusions:
Random sampling alone supports correlation, not causation.
To establish cause-and-effect, you need a controlled experiment with random assignment to treatment groups.
Worked example 1
A poll reports that 62% of respondents support a policy, with a margin of error of ±3%. What is the most defensible interpretation?
Solution
Range. True population support is likely between 62 − 3 = 59% and 62 + 3 = 65%.
The true support is likely between 59% and 65%.
Worked example 2
A researcher wants to know whether a new fertilizer raises yield. Two designs: (i) Survey 500 farms about yield and fertilizer use. (ii) Randomly assign half of her own plots to fertilizer, half to control. Compare yields. Which design supports a causal claim?
Solution
Survey. Design (i) only observes — it can show correlation, but other variables (soil, weather, technique) could explain any pattern.
Experiment. Design (ii) uses random assignment, controlling for confounding variables. This is what justifies causal claims.
Only design (ii) — random assignment in a controlled experiment — supports a cause-and-effect conclusion.
Practice test
8 questions on random sampling, bias, margin of error, and the correlation-vs-causation distinction.
Practice test · 8 questionsQuestion 1 of 8 · Score 0