Inference from Sample Statistics and Margin of Error
Polls, surveys, and scientific samples all try to learn about a huge group by studying a small slice of it. The test wants you to know exactly what conclusions that slice does — and does NOT — let you draw.
Statistic 52% ± 3 gives a plausible range of 49% to 55% for the population.
| Quantity | In population | In sample | |
|---|---|---|---|
| Tagged | 40 | 8 | |
| Total | N (unknown) | 100 | |
| Set up | 40 / N | = 8 / 100 → N = 500 |
Match the tagged-to-total proportion to solve for the unknown population N.
Finally, watch for valid conclusions. You can only generalize to the population that was randomly sampled. If 500 adults in one city were surveyed, your conclusion applies to that city's adults — not the whole state, not teenagers, not anyone outside the sampling frame.
Check your understanding with a question from this topic:
A biologist tags 40 fish in a lake and releases them. A week later, she catches a sample of 100 fish and finds that 8 are tagged. Using the capture-recapture method, what is the estimated total number of fish in the lake?
Worked examples
A biologist tags 60 turtles in a pond and releases them. Two weeks later she catches 80 turtles and finds 12 are tagged. Using the capture-recapture method, what is the best estimate of the total number of turtles in the pond?
A poll of 1,200 likely voters found that 47% plan to vote for Proposition 5, with a margin of error of ±3 percentage points at a 95% confidence level. Which is the best interpretation?
Two surveys estimate the proportion of a city's residents who recycle. Survey 1 sampled 400 residents; Survey 2 sampled 1,600 residents. Both used random sampling. Assuming similar results, which statement is best supported?
Common pitfalls
The margin of error describes uncertainty about the true population value. Saying '44–50% of the people surveyed' is wrong — within the sample, the percentage is exactly the reported statistic.
Margin of error only measures random sampling error. If the sample wasn't randomly drawn from the target population, a larger size just gives you a precise wrong answer.
More data means less uncertainty. Larger sample → smaller margin of error. Don't pick the answer that makes the small sample more precise.
You can only generalize to the population that was actually sampled. A survey of one city's adults says nothing valid about other cities, ages, or the whole country.
Key takeaways
Plausible range = statistic ± margin of error, and it describes the whole population.
Larger random sample → smaller margin of error (it shrinks with 1/√n).
Margin of error measures random error only — it never corrects a biased sample.
Conclusions are valid only for the population that was randomly sampled.
Capture-recapture is a proportion: tagged/total = tagged-in-sample/sample-size.
Watch & learn
Curated Khan Academy walkthroughs on Inference from Sample Statistics and Margin of Error. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.
Try it yourself
5 practice questions on Inference from Sample Statistics and Margin of Error, drawn from the question bank. The tutor is one click away if you get stuck.