Symmetric, skewed left, skewed right, uniform; number of peaks (modes)

Gaps, clusters, and outliers (flag with the 1.5 × IQR rule)

Mean for symmetric data; median for skewed data or data with outliers

Standard deviation with the mean; IQR or range with the median

Every possible group of n has an equal chance of selection

Divide into homogeneous strata, then take an SRS within each

Divide into clusters, randomly pick whole clusters, sample everyone in them

Pick a random start, then take every kth individual

Convenience / voluntary

Easy to gather but prone to bias — avoid for inference

Coef / Estimate (slope row)

The sample slope b — predicted change in y per unit x

The standard error of the slope, SEᵦ

b ÷ SEᵦ, used to test H₀: β = 0

Two-sided probability for the slope t statistic

Standard deviation of the residuals (typical prediction error)

How long is the AP Statistics exam?

The AP Statistics exam is 3 hours long — 90 minutes for the 40 multiple-choice questions and 90 minutes for the 6 free-response questions. A graphing calculator with statistics capability is allowed on both sections.

How is the AP Statistics exam scored?

Multiple choice and free response each count for 50% of the composite score, which is then converted to the AP 1–5 scale. A score of 3 or higher is generally considered passing and qualifying for college credit, though each college sets its own credit policy.

What units are on the AP Statistics exam?

Nine College Board units: (1) Exploring One-Variable Data, (2) Exploring Two-Variable Data, (3) Collecting Data, (4) Probability, Random Variables and Probability Distributions, (5) Sampling Distributions, (6) Inference for Categorical Data: Proportions, (7) Inference for Quantitative Data: Means, (8) Inference for Categorical Data: Chi-Square, and (9) Inference for Quantitative Data: Slopes.

Which units are weighted most heavily on AP Statistics?

On the multiple-choice section, Unit 1 (Exploring One-Variable Data) is 15–23% and Unit 4 (Probability and Distributions) is 10–20% — the two largest. The inference units (6 at 12–15% and 7 at 10–18%) are also major. Chi-square (Unit 8) and slopes (Unit 9) are 2–5% each.

Can I use a calculator on the AP Statistics exam?

Yes. A graphing calculator with statistics capability is allowed and expected on both sections. You should be fluent with one-variable statistics, regression, normal and binomial calculations, and the built-in inference procedures before test day.

Is AP Statistics hard?

AP Statistics rewards conceptual understanding and clear communication more than heavy computation. Many students find the free-response section challenging because it requires showing reasoning, checking conditions, and stating conclusions in context — skills this guide's checkpoints and worked scenarios build directly.

How should I use this AP Statistics study guide?

Work through the nine units in order — data exploration, then data collection, then probability and sampling distributions, then the four inference units. After each unit take the checkpoint quiz to find gaps, then drill that unit with our free practice questions and flashcards before exam day.

Is this AP Statistics study guide free?

Yes — the full guide, the checkpoints, the glossary, the practice questions, and the flashcards are 100% free, with no account required.

How do you describe a distribution in statistics?

Describe a distribution by its shape, center, spread, and any unusual features (outliers or gaps) — remembered as SOCS. Shape is symmetric or skewed; center is the mean or median; spread is the range, IQR, or standard deviation. Always describe it in the context of the variable. This is the core Unit 1 skill. For a roughly symmetric distribution, use the mean and standard deviation; for a skewed distribution or one with outliers, use the median and IQR, which resist extreme values. An outlier is often flagged by the 1.5 × IQR rule: below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

What is a z-score and how do you calculate it?

A z-score measures how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ. A positive z is above the mean, a negative z below. Z-scores standardize different distributions so values can be compared on a common scale. Standardizing is central to Unit 1 and to every later normal calculation. In a normal model, a z-score maps directly to a percentile. The empirical rule follows from z-scores: about 68%, 95%, and 99.7% of a normal distribution lie within z = ±1, ±2, and ±3.

What does the correlation coefficient r tell you?

The correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables. It ranges from −1 to 1: near ±1 means a strong linear pattern, near 0 means weak or no linear pattern. The sign matches the slope's direction. r is a Unit 2 staple. It only measures LINEAR association, so a strong curved relationship can still have an r near 0. Correlation is not causation, and r has no units; r² (the coefficient of determination) gives the fraction of variation in y explained by the linear model.

How do you interpret the slope of a least-squares regression line?

The slope is the predicted change in the response variable y for each one-unit increase in the explanatory variable x. Interpret it in context with units: for every additional unit of x, the model predicts y changes by the slope amount, on average. Always say 'predicted' and use the variable names and units (Unit 2). A residual is the actual value minus the predicted value (observed − predicted). A curved or fanning residual plot signals that a straight-line model is not appropriate for the data.

What is the difference between an observational study and an experiment?

In an observational study you measure variables without imposing a treatment, so you can show association but not causation. In an experiment you deliberately assign treatments, and random assignment lets you draw cause-and-effect conclusions. Only a well-designed experiment can establish causation. This Unit 3 distinction drives what conclusions are allowed. Random SAMPLING (from a population) lets you generalize; random ASSIGNMENT (to treatments) lets you infer causation. Confounding — a lurking variable tied to both the explanatory variable and the response — is the reason observational studies can't prove cause and effect.

What is a simple random sample (SRS)?

A simple random sample of size n is chosen so that every possible group of n individuals in the population has an equal chance of being selected. An SRS avoids bias from the selection process and is the baseline sampling method on which most inference procedures are built. Sampling methods are central to Unit 3. Stratified sampling divides the population into homogeneous groups and samples within each; cluster sampling selects whole groups. Bias — such as voluntary-response or undercoverage bias — is a flaw in the method that pushes estimates consistently away from the truth, and a larger sample cannot fix it.

What is the difference between mutually exclusive and independent events?

Mutually exclusive (disjoint) events cannot happen at the same time, so P(A and B) = 0. Independent events do not affect each other's probability, so P(A and B) = P(A)·P(B). Two events that are mutually exclusive are generally NOT independent — if one happens, the other can't. These probability rules anchor Unit 4. The general addition rule is P(A or B) = P(A) + P(B) − P(A and B). Conditional probability, P(A | B) = P(A and B) ÷ P(B), is how you test independence: A and B are independent when P(A | B) = P(A).

What is the expected value of a random variable?

The expected value (mean) of a discrete random variable is the long-run average outcome: μ = Σ x·P(x), summing each value times its probability. It is the center of the probability distribution — the average you would approach over many, many repetitions. Expected value is a Unit 4 idea. A binomial random variable, counting successes in n independent trials each with probability p, has mean np and standard deviation √(np(1 − p)). A geometric random variable counts the number of trials until the first success and has mean 1 ÷ p.

What is a sampling distribution?

A sampling distribution is the distribution of a statistic — like the sample mean x̄ or sample proportion p̂ — over all possible random samples of a given size from a population. It describes how the statistic varies from sample to sample, which is what makes inference possible. This Unit 5 idea bridges probability and inference. The standard deviation of a sampling distribution is called the standard error; for x̄ it is σ ÷ √n, so larger samples give a tighter distribution. A statistic is an unbiased estimator when the mean of its sampling distribution equals the true parameter.

What is the central limit theorem?

The central limit theorem says that for a large enough sample size, the sampling distribution of the sample mean x̄ is approximately normal — no matter the shape of the population. A common rule of thumb is n ≥ 30. It is why normal-based inference works even on skewed data. The CLT is the cornerstone of Unit 5 and all later inference. The larger the sample, the more normal the sampling distribution becomes. For proportions, the parallel condition is the Large Counts rule: np ≥ 10 and n(1 − p) ≥ 10 for the sampling distribution of p̂ to be approximately normal.

How do you interpret a confidence interval?

A confidence interval gives a range of plausible values for a population parameter. A correct interpretation is: 'We are 95% confident that the true parameter lies between the lower and upper bounds.' The 95% refers to the method — 95% of such intervals capture the true value. Interpreting intervals correctly is heavily tested across Units 6 and 7. The confidence LEVEL describes the long-run capture rate of the procedure, not the probability that one specific interval contains the parameter. An interval is the estimate ± margin of error, where the margin of error = critical value × standard error.

What is a p-value in a significance test?

A p-value is the probability of getting a sample result at least as extreme as the one observed, assuming the null hypothesis is true. A small p-value (often ≤ α = 0.05) means the data would be unlikely under H₀, giving evidence against it, so you reject the null. The p-value is the engine of every significance test (Units 6–9). If p ≤ α, reject H₀; if p > α, fail to reject H₀ — you never 'accept' the null. A p-value is NOT the probability that the null hypothesis is true; it is conditional on the null being true.

What is the difference between a Type I and a Type II error?

A Type I error is rejecting a true null hypothesis — a false positive; its probability equals the significance level α. A Type II error is failing to reject a false null hypothesis — a false negative; its probability is β. The power of a test, 1 − β, is its ability to detect a true effect. Errors and power span Units 6–7. Lowering α reduces Type I errors but raises Type II errors, all else equal. Power increases with a larger sample size, a larger true effect, or a larger α — making the test better at detecting a real difference.

When do you use a t-distribution instead of a z-distribution?

Use a t-distribution for inference about a population MEAN when the population standard deviation σ is unknown and you estimate it with the sample standard deviation s — which is almost always. The t-distribution is bell-shaped but has heavier tails, controlled by its degrees of freedom (n − 1). This is the key Unit 7 decision: z for proportions, t for means. As the sample size grows, the t-distribution approaches the standard normal. Conditions for one-sample t procedures: a random sample, the 10% condition for independence, and an approximately normal population or a large enough n by the CLT.

When do you use a matched-pairs t-test?

Use a matched-pairs t-test when data come in naturally linked pairs — before-and-after measurements on the same subject, or twins, or matched units. You analyze the single column of differences with a one-sample t procedure, testing whether the mean difference μ_d differs from zero. Recognizing paired data is a common Unit 7 skill — the giveaway is two measurements tied to the same individual or matched unit. A two-sample t-test, by contrast, compares the means of two INDEPENDENT groups, where there is no pairing between observations.

What is a chi-square test and when do you use it?

A chi-square (χ²) test compares observed counts to the counts expected under a null hypothesis, using χ² = Σ (observed − expected)² ÷ expected. Use goodness-of-fit for one categorical variable against a claimed distribution, and a test of independence or homogeneity for two categorical variables. Chi-square procedures are Unit 8. A test of independence uses one sample classified by two variables; a test of homogeneity compares the distribution of one variable across several populations or groups. Conditions: a random sample and all expected counts at least 5. A larger χ² statistic gives a smaller p-value and stronger evidence against the null.

How do you test whether a regression slope is significant?

Test a regression slope with a t-test for the slope β. The null hypothesis H₀: β = 0 says there is no linear relationship between x and y in the population. The t statistic is the sample slope b divided by its standard error, with n − 2 degrees of freedom. Inference for slopes is Unit 9 — the highest-level synthesis on the exam. Rejecting H₀: β = 0 is evidence of a real linear association between the variables. Computer regression output gives the slope, its standard error, the t statistic, and the p-value; the conditions are summarized by the acronym LINER (Linear, Independent, Normal, Equal variance, Random).

A variable that records which group or category an individual falls into (e.g., eye color, brand). Summarized with counts and proportions, not means.

The pattern of values a variable takes — described by its shape, center, spread, and any unusual features (the SOCS summary).

The arithmetic average of a data set; the balance point of the distribution. It is pulled toward the tail of a skewed distribution and toward outliers.

The middle value of an ordered data set. It resists outliers, so it is preferred for skewed distributions.

A measure of the typical distance of values from the mean. A larger standard deviation means more spread.

Interquartile range (IQR)

The spread of the middle 50% of the data: Q3 − Q1. Used with the median and resistant to outliers.

A value far from the rest of the data, often flagged by the 1.5 × IQR rule: below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

How many standard deviations a value is from the mean: z = (x − μ) ÷ σ. Standardizes values for comparison.

For a normal distribution, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.

Correlation coefficient (r)

A number from −1 to 1 measuring the strength and direction of a linear relationship between two quantitative variables.

FREE AP Statistics Study Guide 2026 + Readiness Score

This free AP Statistics study guide teaches to the College Board AP Statistics course — all nine content units, organized the way the exam is built.^[1] AP Statistics is a college-level introductory statistics course that builds from exploring data to statistical inference, and the exam rewards clear reasoning and communication as much as computation.^[2]

The guide covers the currentexam format: a 40-question multiple-choice section and a 6-question free-response section, each worth half your score, with a graphing calculator allowed throughout. It’s interactive, not a wall of text: every unit has a built-in checkpoint quiz, hover-able glossary terms, worked statistical examples, and concept questions, so you learn by doing.

Read the guide unit by unit, test yourself at each checkpoint, then round out your free AP Statistics prep with our practice questions and flashcards.

AP Statistics is one of the 17 AP exams — explore our AP study guides to compare and prep across the whole family.

AP Statistics Exam Snapshot

AP Statistics exam at a glance (2026)
Detail	AP Statistics
Questions	46 total — 40 multiple choice + 6 free response
Section I	40 multiple-choice questions · 90 minutes · 50% of score
Section II	6 free-response questions · 90 minutes · 50% of score (Q6 = Investigative Task)
Total time	3 hours
Score scale	1–5; a 3 or higher generally earns college credit
Calculator	Graphing calculator with statistics capability allowed on both sections
Content	9 College Board units, data exploration → inference
When / who	Each May; high school students taking the AP Statistics course
Publisher	College Board

How the AP Statistics exam is built — two equally weighted sections

Multiple choice and free response each count for half of your score. A graphing calculator with statistics capability is expected on both sections.

Section I — Multiple Choice40 questions · 90 minutes · 50% of the score. A graphing calculator is allowed throughout.
Section II — Free Response6 questions · 90 minutes · 50% of the score. You show statistical reasoning and communicate conclusions in writing.
FRQ 1–5 — Short & focusedFive questions, each targeting specific skills and units (data analysis, probability, inference, etc.).
FRQ 6 — Investigative TaskThe longest, most heavily weighted free-response question. It extends a familiar idea into a new, unfamiliar setting and rewards complete statistical communication.

3 hours total · scored 1–5 · a 3 or higher generally earns college credit. Both sections together set the composite that maps to your AP score.

Because the multiple-choice and free-response sections are weighted equally, you can’t coast on one and ignore the other.^[3] Spend your study time across all nine units, but know that data exploration (Units 1–2) and the inference units (5–9) carry the most weight on the multiple-choice section:

AP Statistics multiple-choice weighting by unit (2026 official ranges)

U1 · Exploring One-Variable Data19% · 15–23% of MCQ

U4 · Probability & Distributions15% · 10–20% of MCQ

U7 · Inference: Means14% · 10–18% of MCQ

U3 · Collecting Data13.5% · 12–15% of MCQ

U6 · Inference: Proportions13.5% · 12–15% of MCQ

U5 · Sampling Distributions9.5% · 7–12% of MCQ

U2 · Exploring Two-Variable Data6% · 5–7% of MCQ

U8 · Inference: Chi-Square3.5% · 2–5% of MCQ

U9 · Inference: Slopes3.5% · 2–5% of MCQ

College Board reports each unit’s share as an approximate range, so the exact mix shifts slightly each year.^[1] This guide teaches all nine units in the order the course builds them — exploring data, collecting data, probability and sampling distributions, then the four inference units.

1 · Exploring One-Variable Data

15–23% of the multiple-choice section — the single largest unit. Displaying and summarizing a single variable: graphs, measures of center and spread, outliers, and the normal model.^[1]

Graphs & Describing Distributions

A is summarized with counts and proportions (bar charts, two-way tables); a with dotplots, stemplots, histograms, and boxplots. Describe any with SOCS — shape (symmetric or skewed), center, spread, and unusual features — always in context.

Describing a distribution — the SOCS checklist
Feature	What to report
Shape	Symmetric, skewed left, skewed right, uniform; number of peaks (modes)
Outliers / unusual	Gaps, clusters, and outliers (flag with the 1.5 × IQR rule)
Center	Mean for symmetric data; median for skewed data or data with outliers
Spread	Standard deviation with the mean; IQR or range with the median

Center, Spread & Outliers

The is the balance point and is pulled toward a skew; the resists outliers. Spread is the (with the mean) or the (with the median). An is flagged when a value falls below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$ .

Normal Distributions & z-Scores

A standardizes a value: $z = \dfrac{x - \mu}{\sigma}$ — how many standard deviations it sits from the mean. In a normal model, the gives the 68–95–99.7 percentages, and a z-score maps to a percentile.

The normal distribution and the empirical rule (68–95–99.7)

±1σ
≈ 68% of values

±2σ
≈ 95% of values

±3σ
≈ 99.7% of values

The curve is symmetric about the mean μ. A z-score, z = (x − μ) ÷ σ, tells you how many standard deviations a value sits from the mean.

Checkpoint · Unit 1 · Exploring One-Variable Data

Question 1 of 10

Which measure of center is most resistant to the influence of extreme values in a data set?

2 · Exploring Two-Variable Data

5–7% of the multiple-choice section. The relationship between two variables — scatterplots, correlation, the least-squares line, residuals, and predictions.^[1]

Scatterplots & Correlation

Plot two quantitative variables on a scatterplot and describe the association by direction, form, and strength. The measures the strength and direction of a linear relationship, from $-1$ to $1$ .

Least-Squares Regression & Residuals

The $\hat{y} = a + bx$ minimizes the sum of squared , where a residual is $y - \hat{y}$ (observed − predicted). The slope b is the predicted change in y per one-unit increase in x. The $r^2$ is the fraction of variation in y explained by the model.

Checkpoint · Unit 2 · Exploring Two-Variable Data

Question 1 of 4

Which type of graph is used to display the relationship between two quantitative variables?

3 · Collecting Data

12–15% of the multiple-choice section. How data is gathered determines what conclusions are allowed — sampling methods, bias, experimental design, and the difference between association and causation.^[1]

Sampling Methods & Bias

A gives every group of n individuals an equal chance. Other methods include stratified (sample within homogeneous groups), cluster (sample whole groups), and systematic sampling. — voluntary response, undercoverage, nonresponse — is a flaw in the method that a bigger sample can’t fix.

Common sampling methods
Method	How it works
Simple random sample	Every possible group of n has an equal chance of selection
Stratified	Divide into homogeneous strata, then take an SRS within each
Cluster	Divide into clusters, randomly pick whole clusters, sample everyone in them
Systematic	Pick a random start, then take every kth individual
Convenience / voluntary	Easy to gather but prone to bias — avoid for inference

Experiments vs. Observational Studies

An observational study measures variables without intervening, so it can show association but not causation. An experiment imposes treatments; balances out lurking variables and is what allows a cause-and-effect conclusion. is the reason observational data can’t prove causation.

Checkpoint · Unit 3 · Collecting Data

Question 1 of 10

In a simple random sample of size n, what is true about every possible group of n individuals from the population?

4 · Probability, Random Variables & Distributions

10–20% of the multiple-choice section. The rules of probability, conditional probability and independence, random variables and expected value, and the binomial and geometric distributions.^[1]

Probability Rules

The complement rule is $P(A^c) = 1 - P(A)$ ; the general addition rule is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . events have $P(A \cap B) = 0$ ; events satisfy $P(A \cap B) = P(A)P(B)$ .

Random Variables & Expected Value

A assigns a number to each outcome. Its is $\mu_X = \sum x \cdot P(x)$ — the long-run average. Adding a constant shifts the mean; multiplying by a constant scales both mean and standard deviation.

Binomial & Geometric Distributions

A counts successes in $n$ independent trials, each with probability $p$ : mean $np$ , standard deviation $\sqrt{np(1-p)}$ . A geometric distribution counts trials until the first success, with mean $1/p$ .

Key distributions and their means
Distribution	Mean	Standard deviation
Binomial (n, p)	$np$	$\sqrt{np(1-p)}$
Geometric (p)	$1/p$	$\sqrt{(1-p)}/p$
Sample mean $\bar{x}$	$\mu$	$\sigma/\sqrt{n}$
Sample proportion $\hat{p}$	$p$	$\sqrt{p(1-p)/n}$

Checkpoint · Unit 4 · Probability & Distributions

Question 1 of 10

A random variable X represents the number of successes in a fixed number of independent trials, each with the same probability of success. Which type of probability distribution does X follow?

5 · Sampling Distributions

7–12% of the multiple-choice section. The bridge between probability and inference: how a statistic varies from sample to sample, and why that variation is approximately normal for large samples.^[1]

What a Sampling Distribution Is

A is the distribution of a statistic over all possible samples of a fixed size. Its spread is the ; for the sample mean it is $\sigma/\sqrt{n}$ , so larger samples give a tighter distribution. A statistic is unbiased when the mean of its sampling distribution equals the parameter.

Why a sampling distribution makes inference possible

PopulationAny shape — mean μ, standard deviation σ. You usually can't measure all of it.
Take many random samples of size nEach sample gives one statistic — a sample mean x̄ or a sample proportion p̂.
Sampling distribution of x̄Collect those statistics. Its mean equals μ; its spread (standard error) is σ ÷ √n — smaller as n grows.
Central Limit TheoremFor large enough n, the sampling distribution of x̄ is approximately normal — even if the population is skewed.

Larger samples give a narrower, more normal sampling distribution — that is exactly what lets us build confidence intervals and run significance tests.

The Central Limit Theorem

The says that for a large enough sample, the sampling distribution of $\bar{x}$ is approximately normal regardless of the population’s shape — a common threshold is $n \ge 30$ . For proportions, the parallel condition is the Large Counts rule: $np \ge 10$ and $n(1-p) \ge 10$ .

Checkpoint · Unit 5 · Sampling Distributions

Question 1 of 10

What does a sampling distribution describe?

6 · Inference for Proportions

12–15% of the multiple-choice section. Confidence intervals and significance tests for one and two categorical proportions, plus the logic of hypotheses, p-values, errors, and power.^[1]

The four-step inference workflow — State, Plan, Do, Conclude

1StateState the parameter and the hypotheses (or the confidence level you want).
2PlanName the procedure and check its conditions — Random, 10% / Independent, and Large Counts / Normal.
3DoCompute the test statistic (z or t) and the P-value, or the interval. Show the formula.
4ConcludeCompare P to α and state a conclusion in context — reject / fail to reject, or interpret the interval.

FRQ graders reward this exact structure. Conditions in Plan and a conclusion in context in Conclude are where most points are won or lost.

Confidence Intervals for Proportions

A one-proportion z-interval is $\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ . The is the critical value times the standard error. Interpret the as: “We are 95% confident the true proportion lies between the bounds” — the 95% describes the method.

Significance Tests & Errors

A test pits the against the . The is the probability of a result this extreme assuming H₀ is true; if $p \le \alpha$ you reject H₀. A rejects a true null (probability $\alpha$ ); a fails to reject a false null (probability $\beta$ ); and is $1 - \beta$ .

The two error types
	H₀ is actually true	H₀ is actually false
Reject H₀	Type I error (prob. α)	Correct — power (1 − β)
Fail to reject H₀	Correct decision	Type II error (prob. β)

Checkpoint · Unit 6 · Inference for Proportions

Question 1 of 10

When setting up a significance test, what is the role of the null hypothesis?

7 · Inference for Means

10–18% of the multiple-choice section. Confidence intervals and tests for means using the t-distribution — one-sample, matched-pairs, and two-sample procedures.^[1]

The t-Distribution & One-Sample t

Use a for means whenever $\sigma$ is unknown (the usual case), estimating it with the sample standard deviation s. The t-distribution has heavier tails set by its ( $n - 1$ for one sample). A one-sample t-interval is $\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}$ .

Paired & Two-Sample Means

A analyzes the differences within linked pairs (before/after, twins) with a one-sample t procedure on $\mu_d$ . A two-sample t-test compares two independentgroups’ means. The giveaway for pairing is two measurements tied to the same individual or matched unit.

Checkpoint · Unit 7 · Inference for Means

Question 1 of 10

In a one-sample t interval for a population mean, what does the margin of error represent?

8 · Inference for Chi-Square

2–5% of the multiple-choice section. Comparing observed counts to expected counts for categorical data using the chi-square ( $\chi^2$ ) statistic.^[1]

Goodness-of-Fit Test

A chi-square goodness-of-fit test checks whether the distribution of one categorical variable matches a claimed distribution. The is $\chi^2 = \sum \dfrac{(O - E)^2}{E}$ , where O is observed and E is expected. Degrees of freedom equal the number of categories minus 1.

Independence & Homogeneity

For two categorical variables, use a chi-square test of independence (one sample classified two ways) or homogeneity (comparing the distribution across several groups). For a two-way table, degrees of freedom are $(r-1)(c-1)$ , and each expected count is $\dfrac{\text{row total} \times \text{column total}}{\text{grand total}}$ .

9 · Inference for Slopes

2–5% of the multiple-choice section. The capstone unit: inference for the slope of a least-squares regression line, tying together regression (Unit 2) and the inference workflow (Units 6–7).^[1]

t-Test for the Slope

To test whether a linear relationship exists in the population, test $H_0: \beta = 0$ with a t statistic $t = \dfrac{b}{SE_b}$ — the sample slope over its standard error — on $n - 2$ degrees of freedom. Rejecting $H_0$ is evidence of a real linear association. A slope interval is $b \pm t^* \cdot SE_b$ .

Reading Regression Output

Computer output lists each predictor’s coefficient (the slope b), its standard error $SE_b$ , the t statistic, and the p-value. The conditions for slope inference are summarized by — Linear, Independent, Normal residuals, Equal variance, Random.

Reading regression output for slope inference
Output term	What it is
Coef / Estimate (slope row)	The sample slope b — predicted change in y per unit x
SE Coef	The standard error of the slope, SEᵦ
t statistic	b ÷ SEᵦ, used to test H₀: β = 0
P-value	Two-sided probability for the slope t statistic
s	Standard deviation of the residuals (typical prediction error)

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside official College Board practice and our free tools. AP Statistics rewards reasoning and communication, so the highest-leverage practice is working free-response questions and writing complete conclusions in context. When you reach the inference units, lean on the same four-step framework every time, and use the procedure chooser below to match the data to the correct test.

Which inference procedure? Match the data to the test

One categorical variable, one proportion1-proportion z-interval / z-test (Unit 6)

Compare two proportions2-proportion z-interval / z-test (Unit 6)

One quantitative variable, one mean1-sample t-interval / t-test (Unit 7)

Paired (before/after) measurementsMatched-pairs t-test on the differences (Unit 7)

Compare two means (independent groups)2-sample t-interval / t-test (Unit 7)

Counts across categories of one variableχ² goodness-of-fit test (Unit 8)

Association between two categorical variablesχ² test of independence / homogeneity (Unit 8)

Slope of a regression linet-test / t-interval for the slope β (Unit 9)

Use z for proportions, t for means, and χ² for counts. Identifying the right procedure is the first decision on almost every inference FRQ.

A study loop that actually works

1
Read a unit here
Work through one unit at a time — data exploration first, then collecting data, probability, sampling distributions, and the inference units.
2
Take the checkpoint
The quick check at the end of each unit exposes what didn't stick.
3
Drill the gaps
Send your weak unit straight into the free practice questions and flashcards.
4
Work full, timed FRQs
Practice free-response questions under time, then score them against the rubric — checking conditions and conclusions in context.

AP Statistics Concept Questions

Common AP Statistics ideas the exam actually measures — at least one per unit. Tap any card for a short, exam-ready answer backed by an official source (College Board), then test yourself on them as flashcards.

AP Statistics Glossary

Quick definitions for the terms you’ll see most across AP Statistics:

Alternative hypothesis (Hₐ): The claim a researcher is gathering evidence for — that there is an effect, difference, or relationship.
Bias: A systematic tendency for a sampling or measurement method to favor certain outcomes; a larger sample cannot remove bias.
Binomial distribution: The distribution of the number of successes in n independent trials, each with success probability p; mean np, standard deviation √(np(1 − p)).
Categorical variable: A variable that records which group or category an individual falls into (e.g., eye color, brand). Summarized with counts and proportions, not means.
Central limit theorem (CLT): For large enough samples, the sampling distribution of the sample mean is approximately normal regardless of the population's shape.
Chi-square (χ²) statistic: A measure of how far observed counts fall from expected counts: χ² = Σ (observed − expected)² ÷ expected.
Coefficient of determination (r²): The fraction of the variation in the response variable that is explained by the linear regression model.
Confidence interval: An interval estimate of a parameter, estimate ± margin of error, with a stated confidence level (e.g., 95%).
Confounding: When a lurking variable is associated with both the explanatory variable and the response, so their effects cannot be separated.
Correlation coefficient (r): A number from −1 to 1 measuring the strength and direction of a linear relationship between two quantitative variables.
Degrees of freedom: A parameter (often n − 1) that controls the exact shape of a t- or chi-square distribution.
Distribution: The pattern of values a variable takes — described by its shape, center, spread, and any unusual features (the SOCS summary).
Empirical rule: For a normal distribution, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.
Expected value: The long-run average (mean) of a random variable: μ = Σ x·P(x).
Independent events: Events for which one occurring does not change the probability of the other, so P(A and B) = P(A)·P(B).
Interquartile range (IQR): The spread of the middle 50% of the data: Q3 − Q1. Used with the median and resistant to outliers.
Least-squares regression line: The line ŷ = a + bx that minimizes the sum of squared residuals; used to predict y from x.
LINER conditions: The conditions for slope inference: Linear, Independent, Normal residuals, Equal variance, Random.
Margin of error: The critical value times the standard error; the half-width of a confidence interval.
Matched-pairs design: A design that analyzes the differences within naturally linked pairs of observations using a one-sample t procedure.
Mean: The arithmetic average of a data set; the balance point of the distribution. It is pulled toward the tail of a skewed distribution and toward outliers.
Median: The middle value of an ordered data set. It resists outliers, so it is preferred for skewed distributions.
Mutually exclusive (disjoint): Events that cannot occur at the same time, so P(A and B) = 0.
Null hypothesis (H₀): The default claim of no effect or no difference that a significance test attempts to find evidence against.
Outlier: A value far from the rest of the data, often flagged by the 1.5 × IQR rule: below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
p-value: The probability of a result at least as extreme as the one observed, assuming the null hypothesis is true.
Power: The probability a test correctly rejects a false null hypothesis: 1 − β.
Quantitative variable: A variable that takes numerical values for which arithmetic makes sense (e.g., height, time). Summarized with measures of center and spread.
Random assignment: Using chance to assign experimental units to treatments; it is what allows an experiment to establish cause and effect.
Random variable: A variable whose numerical value is determined by the outcome of a random process.
Residual: The difference between an observed value and the value predicted by the model: residual = observed − predicted.
Sampling distribution: The distribution of a statistic (like x̄ or p̂) over all possible samples of a fixed size from a population.
Significance level (α): The threshold for the p-value below which the null hypothesis is rejected; it equals the probability of a Type I error.
Simple random sample (SRS): A sample chosen so every possible group of n individuals has an equal chance of being selected.
Standard deviation: A measure of the typical distance of values from the mean. A larger standard deviation means more spread.
Standard error: The standard deviation of a sampling distribution; for the sample mean it is σ ÷ √n.
t-distribution: A bell-shaped distribution with heavier tails than the normal, used for inference about means when σ is unknown; shape set by degrees of freedom.
Type I error: Rejecting a true null hypothesis (a false positive). Its probability is α.
Type II error: Failing to reject a false null hypothesis (a false negative). Its probability is β.
z-score: How many standard deviations a value is from the mean: z = (x − μ) ÷ σ. Standardizes values for comparison.

Free AP Statistics Study Materials & Resources

Everything you need to prepare for AP Statistics is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free AP Statistics study materials for active recall, timed practice, and last-minute review:

AP Statistics Practice Test — exam-style questions across all nine units, with explanations.
AP Statistics Flashcards — active-recall decks for the high-yield formulas, distributions, and inference procedures.

AP Statistics Study Guide FAQ

The AP Statistics exam has 46 questions: 40 multiple-choice questions in Section I (90 minutes, 50% of the score) and 6 free-response questions in Section II (90 minutes, 50% of the score). The sixth free-response question is the longer Investigative Task.

References

1.College Board. “AP Statistics Course and Exam Description.” AP Central. ↑
2.College Board. “AP Statistics — AP Students.” College Board. ↑
3.College Board. “AP Statistics Exam — AP Students.” College Board. ↑
4.College Board. “About the AP Statistics Course — AP Central.” AP Central. ↑

Sources for the concept answers

Every answer in the AP Statistics concept questions above is drawn from an official primary source:

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