Educerie · SAT · Math
Problem-Solving and Data Analysis · PSDA.6 Inference from sample statistics and margin of error
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Use a margin of error to give a plausible range | 42% ± 3% is 39% to 45% |
| Say what a larger sample does | Narrows the margin |
| Say what the margin does not cover | Bias, bad questions, dishonest answers |
| Judge whether a sample supports a generalisation | Was it randomly selected, and from what? |
| Compare two intervals | Overlapping ranges support no claim of difference |
1The idea in one paragraph
A sample tells you about the population it was drawn from, and only if it was drawn at random. The margin of error says how much the sample figure might differ from the true population figure by chance alone. It measures luck, not honesty and not method — a badly designed survey of a million people has a tiny margin of error and is still worthless.
Random selection is what licenses the generalisation. The margin of error is only meaningful once that has happened.
2Reading a margin of error
In a random sample of 600 residents, 42% supported the scheme, with a margin of error of 3%.
The plausible range for the whole population is 39% to 45%. The right phrasing, and the one the correct answer uses, is that the true value is plausibly in that interval — not that it is certainly there, and not that 42% of the population supports it.
The options on this question type differ by exactly that hedge, so read them to the end.
3What changes the margin
| Change | Effect on the margin |
|---|---|
| Larger sample | Smaller |
| Smaller sample | Larger |
| Higher confidence level | Larger |
| A better-worded question | No effect |
| A more representative sample | No effect |
The last two matter. The margin of error is computed from the sample size and the confidence level, and it knows nothing about whether the sample was any good.
4What it cannot fix
- A sample that is not random. Surveying people leaving a gym about exercise gives a useless figure however many are asked.
- Leading questions. Wording changes answers, and the margin does not notice.
- Non-response. If the people who ignore a survey differ from those who answer it, the sample is biased.
A question asking "what is the most serious limitation of this study" is usually pointing at one of those three, and the answer will never be the sample size when selection was not random.
5Comparing two intervals
The right conclusion is that the survey does not establish a difference between the groups.
That is the whole question type: overlapping intervals support no claim of difference, and non-overlapping ones support one.
6Generalising to the right population
A random sample of members of one club generalises to that club, not to the town. The correct answer names the population that was actually sampled, and the wrong answers stretch it — to the region, to all adults, to next year.
Where points are lost
- Saying the population value is the sample value.
- Treating the margin as covering bias or bad questions.
- Thinking a bigger sample fixes a non-random one.
- Claiming a difference between overlapping intervals.
- Generalising beyond the population sampled.
- Adding the margin to one end only.
Work it right
- Check how the sample was selected, and from what population.
- Build the interval: figure − margin to figure + margin.
- Say the conclusion with a hedge: plausibly, it is reasonable to estimate.
- Keep the conclusion inside the population actually sampled.
- If two intervals are compared, check whether they overlap.
Try it
Q1. In a random sample of 500 adults in a city, 38% said they use the bus weekly, with a margin of error of 4%. Which conclusion is best supported?
A) Exactly 38% of adults in the city use the bus weekly. B) It is plausible that between 34% and 42% of adults in the city use the bus weekly. C) Between 34% and 42% of all adults in the country use the bus weekly. D) The survey proves that most adults do not use the bus.
Q2. Researchers repeat the survey with 2,000 adults, selected the same way. What happens to the margin of error?
A) It increases. B) It decreases. C) It is unchanged. D) It becomes zero.
Q3. A survey of shoppers at one supermarket finds that 71% cook at home most nights. What is the most serious limitation of concluding that 71% of the town cooks at home most nights?
A) The sample size is too small. B) The sample was not randomly selected from the town. C) The margin of error was not reported. D) The question was about cooking rather than shopping.
Q4. Two random samples give 46% ± 5% and 53% ± 5%. What is the best conclusion?
A) The second group's true percentage is higher. B) The two groups have equal percentages. C) The data do not establish a difference between the groups. D) The margins of error are too small to compare.
Q5. A random sample of 800 members of a sports club is surveyed. To which population can the results be generalised?
A) All residents of the city B) All members of that sports club C) All people who play sport D) All adults in the country
In one breath
A margin of error measures chance alone: it tells you how far the sample figure might sit from the population figure by luck, and it knows nothing about a leading question, a biased sample or people who refused to answer. Build the interval, state the conclusion with a hedge, and keep it inside the population that was actually sampled — and when two intervals overlap, the honest conclusion is that no difference has been established.
Answers
Q1. B. the interval, hedged, and the right population
A treats the sample as exact. C stretches a city sample to the country. D overreaches into a claim the numbers do not make.
Q2. B — it decreases. a larger sample narrows the margin D is the trap: no finite sample removes uncertainty entirely.
Q3. B. selection, not size Supermarket shoppers are not a random sample of the town, and no sample size repairs that. A is the answer students reach for first; C and D describe real gaps that matter far less.
Q4. C. the intervals overlap
A claims a difference the data cannot support; B claims equality, which is equally unsupported.
Q5. B. a sample generalises to the population it was drawn from The others all stretch beyond the club.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Problem-Solving and Data Analysis, skill/knowledge testing point "Inference from sample statistics and margin of error"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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