Math & Computer Science σSTATS

AP Statistics

Exploring data, designing studies, and using probability to draw statistical conclusions.

Study guide last reviewed September 25, 2026 · see what changed

Exam Format

Format reflects the current published structure — always confirm against AP Central before test day, since formats occasionally change.

SectionDetailsTimeWeight
Section I – Multiple Choice40 questions1 hr 30 min50%
Section II – Free Response5 short + 1 investigative task1 hr 30 min50%

Frequently Asked Questions

Straight answers about the AP Statistics exam — format, credit, dates, and this site.

What's the format of the AP Statistics exam?

Section I – Multiple Choice: 40 questions (1 hr 30 min, 50% of the total score); Section II – Free Response: 5 short + 1 investigative task (1 hr 30 min, 50% of the total score). See the Exam Format tab above for the full breakdown, and always double-check the current year's format on AP Central before test day.

Is the AP Statistics exam multiple-choice, written response, or portfolio-based?

It's multiple-choice questions and written free-response or essay questions.

What score do I need on AP Statistics for college credit?

It depends entirely on the college — most award credit for a 3 or higher, but many selective schools require a 4 or a 5, and some don't grant credit at all for certain scores. Always check your target college's own AP credit policy page rather than assuming a 3 will count everywhere.

When is the AP Statistics exam?

AP exams are administered during the College Board's annual testing window in the first two full weeks of May, with the exact date and time for each subject set by the College Board and announced the preceding fall. Check AP Central or your school's AP coordinator for this year's exact date.

Is this AP Statistics study guide actually free?

Yes. The unit-by-unit study guide, worked examples, practice quiz, and full-length timed practice exam on this page are all free, with no login, signup, or paywall.

Study Guide

Unit-by-unit breakdown of what the course actually covers, in teaching order.

Exploring One-Variable Data

Describing distribution shape (symmetric, skewed left/right), center (mean vs. median, and when each is more appropriate), and spread (range, IQR, standard deviation), plus the 1.5×IQR rule for identifying outliers and constructing/interpreting boxplots, histograms, and dotplots for a single quantitative variable.

Exploring Two-Variable Data

Scatterplots for visualizing relationships between two quantitative variables, correlation coefficient (r) measuring strength and direction of linear association, least-squares regression line calculation and interpretation (slope and y-intercept in context), residual plots for checking whether a linear model is appropriate, and the distinction between correlation and causation.

Collecting Data

Sampling methods (simple random, stratified, cluster, systematic) and their tradeoffs, sources of bias (undercoverage, nonresponse, response bias), and experimental design principles: control groups, randomization, replication, and blocking, plus the crucial distinction that only well-designed experiments (not observational studies) support causal claims.

Probability, Random Variables & Distributions

Basic probability rules (addition rule, multiplication rule, conditional probability, independence), discrete random variables and their probability distributions, expected value and variance calculations, and named distributions including the binomial (fixed trials, two outcomes) and geometric (trials until first success) distributions with their specific formulas.

Sampling Distributions

The sampling distribution of a sample statistic (like the sample mean or proportion) as the theoretical distribution of that statistic across all possible samples, the Central Limit Theorem guaranteeing approximate normality for large samples regardless of the population's shape, and how standard error (not standard deviation) quantifies sample-to-sample variability.

Inference for Categorical Data: Proportions

One-sample and two-sample z-procedures for proportions: confidence intervals (estimate ± margin of error) and hypothesis tests (null/alternative hypotheses, p-values, and conclusions in context), including checking the required conditions (random, independent/10% condition, large counts) before running any procedure.

Inference for Quantitative Data: Means

One-sample and two-sample t-procedures for means (using the t-distribution since population standard deviation is typically unknown), paired versus unpaired (independent samples) t-tests and when each applies, and the same four-step inference process (hypotheses, conditions, calculations, conclusion) applied to means instead of proportions.

Inference for Categorical Data: Chi-Square

Chi-square goodness-of-fit tests (comparing one categorical variable's observed distribution to an expected/claimed distribution), and chi-square tests of independence/homogeneity (examining association between two categorical variables), with the chi-square statistic formula and degrees of freedom calculated differently for each test type.

Inference for Quantitative Data: Slopes

Inference for the slope of a least-squares regression line, testing whether the true population slope differs significantly from zero (i.e., whether a linear relationship genuinely exists) using a t-test, and constructing confidence intervals for the slope — the course's capstone unit connecting regression from earlier back to formal inference.

Worked Examples

Concrete, fully worked examples — this is what separates a page reviewers see as genuinely useful from one that just lists topics.

Exploring One-Variable Data

Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Find the median, IQR, and check for outliers.

Median = average of the 4th and 5th values = (4+5)/2 = 4.5. Q1 = 4 (median of lower half), Q3 = 6 (median of upper half), so IQR = 2. Outlier fences: 4 − 1.5(2) = 1 and 6 + 1.5(2) = 9 — no value falls outside [1, 9], so there are no outliers (9 sits exactly at the fence, not beyond it).

Exploring Two-Variable Data

Regression: given ŷ = 2 + 3x and r = 0.85, find the residual for the point (4, 20), and interpret r.

Predicted: ŷ = 2 + 3(4) = 14. Residual = observed − predicted = 20 − 14 = 6 (the model underpredicts this point). r = 0.85 shows a strong, positive linear association — but correlation alone never justifies a causal claim.

Collecting Data

Design flaw: a study lets volunteers choose whether they receive a new tutoring program, then compares their test scores to those who opted out.

Since assignment isn't random, this is an observational study, not an experiment — motivated students may have self-selected into tutoring, so any score difference could reflect that pre-existing motivation (a confounding variable) rather than the program itself. Only randomized assignment supports a causal conclusion.

Probability, Random Variables & Distributions

Binomial: X = number of successes in 5 independent trials with p = 0.3. Find P(X = 2) and E(X).

P(X=2) = C(5,2)(0.3)²(0.7)³ = 10(0.09)(0.343) = 0.3087. E(X) = np = 5(0.3) = 1.5.

Inference for Categorical Data: Proportions

One-sample z-test: H₀: p = 0.5, Hᴥ: p ≠ 0.5, sample p̂ = 0.58, n = 150.

SE = √[0.5(0.5)/150] ≈ 0.0408. z = (0.58−0.5)/0.0408 ≈ 1.96, giving a two-sided p-value right around 0.05 — a genuinely borderline result worth flagging explicitly in a conclusion rather than rounding to a confident yes/no.

Inference for Quantitative Data: Means

Paired t-test: before/after weights on 15 subjects give mean difference d̄ = 3.2, sd = 4.1.

SE = 4.1/√15 ≈ 1.059. t = 3.2/1.059 ≈ 3.02, df = 14 — compare against the critical t-value for the stated significance level to decide.

Inference for Categorical Data: Chi-Square

Goodness-of-fit: observed counts 30, 25, 45 against an expected equal split (33.33 each).

χ² = ∑(O−E)²/E = (30−33.33)²/33.33 + (25−33.33)²/33.33 + (45−33.33)²/33.33 ≈ 6.5, with df = 2.

Inference for Quantitative Data: Slopes

Slope inference: a regression gives b₁ = 2.5, SE(b₁) = 0.8, n = 20.

t = 2.5/0.8 = 3.125, df = 18 — testing H₀: β₁ = 0 this way is how you formally decide whether a linear relationship genuinely exists, rather than just eyeballing a scatterplot.

Study Tips

  • Always state hypotheses, check conditions by name, and interpret in context — free-response scoring is checklist-based, not vibes-based.
  • Practice the investigative task under time pressure; it strings together several units in one multi-part problem.
  • Every inference problem needs all four steps — hypotheses/parameter, conditions checked by name, calculation, conclusion in context — and skipping the conditions check is the most common way to lose points even with flawless arithmetic.
  • 'Strong correlation' is never the same as causation — say so explicitly whenever a prompt could tempt a causal claim from observational data.
  • On the investigative task, read the entire multi-part prompt before starting — later parts often build on earlier ones, and misreading the setup costs points throughout the whole question.

Common Mistakes, Ranked by Point Loss

The errors that cost the most points across the whole exam, not just one unit.

  • 1. Skipping or hand-waving the conditions check (random, independence/10%, large counts or normality) before running an inference procedure.
  • 2. Confusing standard deviation (spread of the raw data) with standard error (spread of a sample statistic) — related concepts, but not interchangeable in a formula.
  • 3. Writing a conclusion that 'accepts' the null hypothesis instead of correctly stating you 'fail to reject' it — a hypothesis test never proves the null true.
  • 4. Running a two-sample t-test on data that's actually paired, or vice versa — the study design determines the correct procedure, not just having two columns of numbers.
  • 5. Rounding a p-value or test statistic before comparing it to alpha, which can flip a genuinely borderline decision.

Practice Quiz

Unofficial practice questions written for this study guide — not official AP content and not from College Board. Use them to self-check, then use the real past free-response questions below for authentic exam practice.

Full Exam Simulation

Unofficial practice, timed to feel like the real thing. Section I pulls from this page's full multiple-choice bank; Section II gives you free-response prompts to write out and self-score against a rubric. Bank size is still growing toward full exam length — timing is paced per-question, not pulled from this subject's official duration.

Section I: 15 multiple-choice questions (~22 min) · Section II: 2 free-response prompts (~30 min)
Section I: Multiple Choice --:--