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Math foundations

SAT surveys and margin of error: what a sample can (and can’t) tell you

A poll says 54% support a proposal, with a margin of error. Does a majority really support it? Change the sample size in an interactive lab and learn the inference rules the SAT tests.

SPSAT Practice Editorial8 min read

The short version

What you'll take away

  • A margin of error turns a sample estimate into a range of plausible values for the whole population.
  • Larger random samples give smaller margins: quadrupling the sample size roughly halves the margin.
  • Results generalize only to the population that was randomly sampled, and only random assignment supports cause-and-effect claims.
In this article

An estimate comes with a range

Surveying a whole population is usually impossible, so researchers survey a random sample and use it to estimate the population value. Because a different random sample would give a slightly different result, the estimate comes with a margin of error.
An estimate of 54% with a margin of error of 3 percentage points means the true population value is plausibly between 51% and 57%. On the SAT, read “estimate ± margin” as a range of plausible values, not a single exact answer.

Sample size changes the margin

Suppose a random sample finds that 54% of voters support a proposal. Is that a majority? It depends on the margin of error, which depends on the sample size. Choose a sample size below and watch the range shrink.

Try it yourself · interactive

Same estimate, different sample sizes

Sample size
Margin of error
± 10.0 pts
Plausible range
44.0%–64.0%
Above 50%?
Can’t tell: the range includes 50%
Margins use the common shortcut margin ≈ 1/√n, about 95% confidence for results near 50%. With n = 100, the range includes 50%, so a majority is not established. With n = 1,600, it is.

Who does the result describe?

A random sample supports conclusions about the population it was drawn from, and no other. A random sample of students at one high school describes that school, not all teenagers in the country. A survey of gym members cannot describe all adults in a city.
Samples that are not random, such as volunteers or online polls that anyone can answer, may be biased. People who choose to respond can differ systematically from people who do not.

Association or cause?

The SAT also tests what kind of study supports what kind of claim. The two key questions are whether the participants were randomly selected and whether the treatments were randomly assigned.

See the skill

What each design lets you conclude

Random sample, no assignment

Generalize, no cause

Describes the population, but shows only an association.

Random assignment, volunteers

Cause, limited scope

Supports a cause for people like the participants only.

Random sample and random assignment

Cause and generalize

The strongest design: causal and applies to the population.

Neither

Limited conclusions

Shows an association among the people studied only.

Use the visual as a reasoning aid: tie each choice to the relationship, grammar, or evidence the passage actually gives.

Your turn

Quick practice

A random sample of 500 residents of City A found that 62% support a new park, with a margin of error of 4 percentage points. Which conclusion is most appropriate?

Hint: Check two things: the range and the population.

Choose one answer

Practice: try these before checking the answers

Name the population and the design before answering.
  • 1. An estimate is 40% with a margin of error of 3 points. What range of values is plausible?
  • 2. A survey of gym members finds most exercise daily. Can this be applied to all adults in the city?
  • 3. A sample size grows from 250 to 1,000. What happens to the margin of error, approximately?
  • 4. Volunteers are randomly assigned to two study methods, and group A scores higher. What can be concluded?

Sources & further reading

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