AP Statistics is about interpreting data and drawing conclusions with quantifiable uncertainty. These notes organize the four major themes — exploring data, sampling, probability, and inference — with the specific formulas and conditions tested on the FRQ.
Units 1–2: Exploring Data
Describing distributions: shape (symmetric, skewed left/right, bimodal), center (mean vs. median), spread (standard deviation, IQR), and outliers (1.5·IQR rule). For skewed distributions, median and IQR are more resistant than mean and SD. Two-variable: scatterplot, linear regression ŷ = a + bx where b = r(sy/sx). Residual = observed − predicted; residual plots should show no pattern.
Units 3–5: Probability and Distributions
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Conditional probability: P(A|B) = P(A ∩ B)/P(B). Independence: P(A|B) = P(A). Binomial distribution: P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; μ = np, σ = √(np(1−p)). Normal distribution: standardize z = (x−μ)/σ; use z-table. Geometric: P(X=k) = (1−p)ᵏ⁻¹p.
Unit 6: Sampling Distributions
Central Limit Theorem: sampling distribution of x̄ is approximately normal when n ≥ 30 (or population is normal). Standard error of x̄: σ/√n. Sampling distribution of p̂: mean = p, SE = √(p(1−p)/n). Conditions for inference: random sample, independence (10% rule: n ≤ N/10), normality (n large enough).
Units 7–9: Inference
Confidence interval structure: estimate ± margin of error (critical value × SE). P-value = probability of getting data this extreme given H₀ is true; small p-value (< α) → reject H₀. Type I error = reject true H₀ (probability = α). Type II error = fail to reject false H₀ (probability = β). Chi-square tests: goodness-of-fit and independence; expected = (row total × col total)/n. Regression inference: t-test for slope.
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