SAT Math · Problem-Solving and Data Analysis · Skill 6 of 7

Inference and margin of error

Use a sample to estimate a whole population, turn a margin of error into a range of plausible values, and know when a result applies to everyone and when it does not.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
Start sampling →Jump to the challenge
Hi, I’m Io. You can learn a lot about a big group by asking a few people. The trick is knowing how sure you can be.
Tap Io

Draw your own samples

Explore · 5 min

In a town, exactly 40% of people support a new park, but a pollster does not know that. Each dot below is one random sample: the percent of that sample who said yes. Draw small samples and large samples and compare how spread out the dots are.

Population, sample, parameter, statistic

Learn · 4 min

  • Population: the whole group you want to know about
  • Sample: the part of the population you actually measure
  • Parameter: a number describing the population, usually unknown
  • Statistic: a number calculated from the sample, used to estimate the parameter

A school has 1,200 students. 80 of them are chosen at random and asked how many hours they sleep. Sort each card.

From sample to population

Practise · 5 min

With a random sample, assume the population behaves like the sample. Find the sample fraction, then multiply by the population size.

estimate = (sample count ÷ sample size) × population size
Worked example: a random sample of 200 of a town’s 6,000 households finds 34 own an electric car. Estimate how many households in the town own one.
  1. Sample fraction: 34 ÷ 200 = 0.17
  2. Scale up: 0.17 × 6,000 = 1,020 households
  3. This is an estimate. A margin of error says how far off it might be

Margin of error

Learn · 5 min

A margin of error turns one estimate into a range of plausible values for the population. Values inside the range are plausible. Values outside it are unlikely. Nothing is guaranteed.

plausible range = estimate − margin to estimate + margin

What changes the margin of error?

Learn · 4 min

  • Larger sample: smaller margin of error, as you saw in chapter 1
  • More variability in the data: larger margin of error
  • Higher confidence level, for example 99% instead of 95%: larger margin of error
  • A bigger sample only helps if it is random. A huge biased sample is still biased

Random samples and who the results fit

Learn · 4 min

A sample result can be generalised to a population only if the sample was chosen at random from that population. People who volunteer, or who are easy to reach, can differ in ways that bias the result.

Test-day checkAsk two questions: was the sample random, and from which group was it drawn? The answer to the second is as far as the conclusion can reach.

Comparing two estimates

Practise · 4 min

To compare two groups, write out both plausible ranges. If they overlap, the true values could be equal, so the data do not show a clear difference. If they do not overlap, the data support a real difference.

Inference in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
Scale upSample fraction × population size.
Plausible rangeEstimate ± margin of error.
Plausible, not certainInside the range is plausible. Nothing is guaranteed.
Smaller marginTake a larger random sample.
Who it applies toOnly the population the random sample came from.
ComparingOverlapping ranges show no clear difference.
Challenge