Unit 1: Exploring One-Variable Data
~15-23% of Exam
Big Ideas
This unit is about describing distributions of a single variable using graphs and numbers. Every description needs Shape, Center, Spread, and Unusual Features (outliers/gaps).
Key Vocabulary
Individuals The objects described by data
Variable A characteristic of an individual
Categorical Groups or categories
Quantitative Numerical values
Distribution Pattern of values
Skewed Tail pulls one direction
Graphs You Must Know
- Dotplot: Each dot = one observation. Great for small data sets.
- Histogram: Bars show frequency of intervals. Do NOT put spaces between bars.
- Stemplot: Shows individual values and shape. Include a key.
- Boxplot: Shows five-number summary. Great for comparing distributions.
Measures of Center
Mean (x-bar): Sum of all values / n. Sensitive to outliers (non-resistant).
Median: Middle value when ordered. Resistant to outliers.
Rule: If skewed right, mean > median. If skewed left, mean < median.
Measures of Spread
- Range: Max - Min. Very sensitive to outliers.
- IQR: Q3 - Q1. Resistant to outliers. Used for boxplots.
- Standard Deviation (s): Average distance from the mean. Non-resistant.
Outlier rule: Below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR)
Normal Distributions
- Symmetric, bell-shaped, described by mean (mu) and standard deviation (sigma)
- 68-95-99.7 Rule: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD
- z-score: z = (x - mean) / SD. Tells you how many SDs from the mean.
AP Exam Tip: When asked to "compare" distributions, you MUST use comparison language ("is greater than," "has more variability than"). Don't just describe each one separately.
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Unit 2: Exploring Two-Variable Data
~5-7% of Exam
Big Ideas
This unit is about the relationship between two quantitative variables. The main tools are scatterplots, correlation, and linear regression.
Scatterplots
- Describe: Direction, Form, Strength, and Unusual Features
- Explanatory variable (x-axis) vs. Response variable (y-axis)
Correlation (r)
r measures the strength and direction of a LINEAR relationship.
• Always between -1 and 1
• r has no units; doesn't change if you switch x and y
• Only measures LINEAR association (a curve could have r near 0)
Common Trap: r = 0.7 does NOT mean "70% of the variation is explained." That's r-squared = 0.49 = 49%.
Least Squares Regression Line (LSRL)
y-hat = a + bx
b = r(sy/sx) a = y-bar - b(x-bar)
r² = proportion of variation in y explained by x
- Slope interpretation: "For each additional [unit of x], the predicted [y] increases/decreases by [b] [units of y]."
- y-intercept interpretation: "When [x] is 0, the predicted [y] is [a] [units]." (May not make sense in context!)
- r-squared interpretation: "[r-squared]% of the variability in [y] is explained by the linear relationship with [x]."
Residuals
Residual = Observed - Predicted = y - y-hat
Positive residual = model underestimated. Negative = overestimated.
A good model has a residual plot with NO pattern (random scatter).
AP Exam Tip: ALWAYS interpret slope and r-squared IN CONTEXT. Generic answers like "as x increases, y increases" lose points. Use the variable names from the problem.
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Unit 3: Collecting Data
~12-15% of Exam
Big Ideas
How data is collected determines what conclusions you can draw. This unit is about sampling methods, experiments, and the difference between the two.
Observational Study vs. Experiment
Observational study: Observe without interfering. Can show ASSOCIATION but NOT causation.
Experiment: Deliberately impose treatments. CAN show causation if well-designed.
Sampling Methods
- Simple Random Sample (SRS): Every individual has equal chance; every group of size n has equal chance
- Stratified: Divide into groups (strata), then SRS from each stratum
- Cluster: Divide into clusters, randomly select entire clusters
- Systematic: Select every k-th individual from a list
Sources of Bias
- Undercoverage: Some groups left out of the sampling frame
- Nonresponse: Selected individuals don't participate
- Response bias: Respondents give inaccurate answers (wording, social desirability)
- Voluntary response: People self-select (strong opinions overrepresented)
Experiments: Key Principles
- Random assignment: Subjects randomly assigned to treatment groups (creates equivalent groups)
- Control: A comparison group (often placebo)
- Replication: Enough subjects to detect real differences
- Blinding: Single-blind (subjects don't know) or double-blind (neither subjects nor evaluators know)
Don't Confuse: Random SELECTION (choosing from population → generalizability) vs. Random ASSIGNMENT (assigning to groups → causation). They are different!
AP Exam Tip: When describing an experiment, always specify: (1) treatments, (2) how subjects are randomly assigned, (3) what response variable is measured, and (4) comparison between groups.
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Unit 4: Probability, Random Variables, and Probability Distributions
~10-20% of Exam
Basic Probability Rules
P(A or B) = P(A) + P(B) - P(A and B) [General Addition Rule]
P(A and B) = P(A) × P(B|A) [General Multiplication Rule]
If independent: P(A and B) = P(A) × P(B)
If mutually exclusive: P(A and B) = 0
Conditional Probability
P(A|B) = P(A and B) / P(B)
"The probability of A, given that B has occurred."
Independence check: Events are independent if P(A|B) = P(A).
Random Variables
- Discrete: Countable outcomes (e.g., number of heads)
- Continuous: Any value in an interval (e.g., height)
Mean of X: μx = Σ[xi × P(xi)]
Variance: σ²x = Σ[(xi - μx)² × P(xi)]
Transformations: If Y = aX + b, then:
μY = aμX + b
σY = |a| × σX (adding b does NOT change spread!)
Combining Random Variables
μ(X+Y) = μX + μY
μ(X-Y) = μX - μY
If INDEPENDENT: σ²(X±Y) = σ²X + σ²Y (ALWAYS ADD variances!)
Critical: When subtracting random variables, you still ADD the variances. Variability always increases when combining.
Binomial Distribution
Conditions (BINS): Binary outcomes, Independent trials, Number of trials is fixed, Same probability each trial.
X ~ Binomial(n, p)
μ = np σ = √(np(1-p))
Geometric Distribution
How many trials until the FIRST success?
P(X = k) = (1-p)^(k-1) × p
μ = 1/p
AP Exam Tip: For probability questions, ALWAYS define your events, state the rule you're using, and show your work. Even on MC, write out the setup to avoid errors.
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Unit 5: Sampling Distributions
~7-12% of Exam
Big Ideas
This is the bridge between probability and inference. A sampling distribution shows what happens when you take MANY samples from the same population.
Key Vocabulary
Parameter: A number describing the POPULATION (p, μ, σ). Fixed but usually unknown.
Statistic: A number describing a SAMPLE (p-hat, x-bar, s). Varies from sample to sample.
Sampling Distribution of p-hat (Sample Proportion)
Mean: μ(p-hat) = p
SD: σ(p-hat) = √(p(1-p)/n)
Approximately Normal when: np ≥ 10 AND n(1-p) ≥ 10
Sampling Distribution of x-bar (Sample Mean)
Mean: μ(x-bar) = μ
SD: σ(x-bar) = σ/√n
Normal when: Population is Normal, OR n ≥ 30 (CLT)
Central Limit Theorem (CLT)
For large enough sample sizes (generally n ≥ 30), the sampling distribution of x-bar is approximately Normal, regardless of the shape of the population. This is one of the most important ideas in all of statistics.
Don't Confuse: The SD of the sampling distribution gets SMALLER as n increases. Larger samples = less variability = more precise estimates.
AP Exam Tip: Know the difference between "the distribution of the population," "the distribution of the sample," and "the sampling distribution of the statistic." They are three different things!
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Unit 6: Inference for Categorical Data — Proportions
~12-15% of Exam
Big Ideas
This is the first inference unit. You're using sample data to draw conclusions about population proportions. EVERY inference problem follows the same 4-step framework.
The 4-Step Inference Framework
1. STATE: Define parameter, state hypotheses (for tests) or confidence level (for intervals)
2. PLAN: Name the procedure and check conditions
3. DO: Calculate the test statistic or interval
4. CONCLUDE: Make a conclusion in context
Conditions to Check (Every Time!)
- Random: Data comes from a random sample or random assignment
- 10% Condition: n ≤ (1/10) of population (for independence)
- Large Counts:
- For intervals: np-hat ≥ 10 AND n(1-p-hat) ≥ 10
- For tests: np0 ≥ 10 AND n(1-p0) ≥ 10 (use the null hypothesis value!)
Confidence Interval for One Proportion
p-hat ± z* × √(p-hat(1-p-hat)/n)
z* values: 90% → 1.645 | 95% → 1.96 | 99% → 2.576
Correct interpretation: "We are [C]% confident that the true proportion of [context] is between [lower] and [upper]."
WRONG: "There is a [C]% probability that the true proportion is in this interval." (The interval either contains p or it doesn't.)
Hypothesis Test for One Proportion (z-test)
z = (p-hat - p0) / √(p0(1-p0)/n)
Use p0 (null hypothesis value) in the standard error!
Two-Proportion z-test and z-interval
Compare two groups. For the test, use combined (pooled) p-hat. For the interval, use separate p-hats.
p-value Interpretation
"Assuming [null hypothesis is true], the probability of observing a result as extreme as or more extreme than the one obtained is [p-value]."
AP Exam Tip: NEVER say "accept the null." Say "fail to reject." NEVER say "the null is true/false." You either have or don't have convincing evidence against it.
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Unit 7: Inference for Quantitative Data — Means
~12-15% of Exam
Big Ideas
Same 4-step framework as Unit 6, but now for means instead of proportions. We use the t-distribution because we don't know the population standard deviation.
Why t instead of z?
We estimate σ with s (sample standard deviation). The t-distribution accounts for this extra uncertainty. It's wider than the Normal distribution, especially for small n. As n increases, t approaches z.
Conditions (Same spirit, different details)
- Random: Random sample or random assignment
- 10% Condition: n ≤ (1/10) of population
- Normal/Large Sample: Population is Normal, OR n ≥ 30, OR n < 30 but no strong skewness or outliers in the sample
One-Sample t-interval
x-bar ± t* × (s / √n)
df = n - 1
One-Sample t-test
t = (x-bar - μ0) / (s / √n)
df = n - 1
Matched Pairs
When two measurements are taken on the SAME individual (or matched individuals), compute the differences first, then do a one-sample t-procedure on the differences. d-bar = mean of differences, sd = SD of differences.
Two-Sample t-procedures
- Compare means from two INDEPENDENT groups
- Check conditions for BOTH samples separately
- Use calculator for df (it's messy)
- DO NOT pool! (AP Stats does not use pooled t-procedures)
AP Exam Tip: On the FRQ, if you're asked to "justify" using a t-procedure, you need to address the Normal/Large Sample condition specifically. "n = 50 > 30, so by CLT the sampling distribution is approximately Normal" is a good response.
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Unit 8: Inference for Categorical Data — Chi-Square
~2-5% of Exam
Three Types of Chi-Square Tests
Goodness of Fit (GOF): One variable. Does the distribution match a claimed/expected distribution?
Independence: One sample, two variables. Are the variables associated?
Homogeneity: Multiple populations/groups, one variable. Same distribution across groups?
The Chi-Square Statistic
χ² = Σ (observed - expected)² / expected
Expected count (two-way table) = (row total × column total) / grand total
Conditions
- Random: Random sample or random assignment
- 10% Condition: n ≤ 1/10 of population
- Large Counts: ALL expected counts ≥ 5
Key Details
- Chi-square tests are ALWAYS right-tailed (large χ² = big discrepancy = evidence against H0)
- df for GOF = (number of categories - 1)
- df for Independence/Homogeneity = (rows - 1)(columns - 1)
- If you reject H0, do a follow-up: which cells contributed most to χ²?
Don't Confuse: Independence and Homogeneity use the SAME formula and same mechanics. The difference is in the study design: one sample vs. multiple populations.
AP Exam Tip: When writing hypotheses for chi-square: GOF compares to specific proportions. Independence says "there is no association between [X] and [Y]." Homogeneity says "the distribution of [X] is the same across [groups]."
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Unit 9: Inference for Quantitative Data — Slopes
~2-5% of Exam
Big Ideas
Use the LSRL from Unit 2, but now ask: is the linear relationship statistically significant? This is inference on the slope (β) of the population regression line.
Hypotheses
H0: β = 0 (No linear relationship in the population)
Ha: β ≠ 0 (or > 0, or < 0, depending on context)
Conditions (LINER)
- Linear: Scatterplot and residual plot show linear pattern
- Independent: Observations are independent (10% condition or study design)
- Normal: Residuals are approximately Normal (check histogram or Normal probability plot of residuals)
- Equal variance: Residual plot shows consistent spread (no fan/cone shape)
- Random: Data comes from a random sample or randomized experiment
t-test for Slope
t = (b - 0) / SE(b)
df = n - 2
Confidence Interval for Slope
b ± t* × SE(b)
df = n - 2
Reading Computer Output
You MUST be able to read a regression output table. Key columns:
• Coef (or Estimate): b = slope, a = intercept
• SE Coef (or Std Error): standard error of the slope
• T (or t-value): test statistic
• P (or p-value): for testing H0: β = 0
• S = standard deviation of residuals
• R-sq = r-squared
AP Exam Tip: The FRQ often gives you computer output and asks you to conduct a full test for the slope. Practice reading output tables! Know where to find b, SE(b), t, p, s, and r-squared. Remember to check all five LINER conditions.
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