AP Statistics / Inference for Quantitative Data: Means
Lesson
Means, t-procedures, and conditions
Use the sample mean and a t critical value when the population standard deviation is unknown.
Learning goals
- Explain why t is used for means when s replaces σ.
- Check conditions for a one-sample t interval.
Explanation
The sample mean is a statistic. Across random samples it centers at the population mean, and its spread shrinks like σ/√n. In practice you almost never know σ, so you plug in s. That extra uncertainty is why t-distributions have heavier tails than the normal curve.
Degrees of freedom are n − 1 for one sample. Conditions still matter: random data, independence, and a distribution that is not wildly skewed when n is small. A quick plot is part of the procedure, not decoration.
Key terms
- t-distribution. A family of bell-shaped models with heavier tails than the normal curve, used when spread is estimated from the sample.
- Degrees of freedom. A value that sets which t-curve you use; for one sample it is n − 1.
Common mistakes
- Using z with s just because n looks “pretty big.”
- Ignoring a strong outlier in a sample of 8 measurements.
Practice
Original Marlow Works items. Check the answer explanation after you try.
A reporter stands at one grocery-store exit and asks shoppers whether they support a new park tax. Why is this a poor way to estimate citywide support?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Why is a 95% t-interval for n = 8 wider than a 95% z-interval with the same s?
Answer. The t critical value is larger than 1.96 when df is small, so the margin of error grows.
Estimating spread from a tiny sample adds uncertainty the z shortcut pretends is gone.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Statistics course page
Official revised 2026–27 five-unit framework.
College Board · All rights reserved · accessed 2026-10-01
Open sourceCC BY — attribution required
Collaborative Statistics
Open statistics text for distributions, probability, and inference.
Open Textbook Library listing · CC BY · accessed 2026-10-01
Open source