AP Statistics: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
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A statistical investigation starts with a question about a group and anticipates variability among its members. The observational unit is the person or object measured; a variable records one characteristic of that unit.
Describe center and position
A distribution describes which values occur and how often. Center, spread, shape and unusual observations tell complementary parts of its story. A single average cannot replace the distribution.
Spread and normal models
Standard deviation measures typical dispersion around a mean in the original measurement units. A standardized score expresses a value in units of standard deviation. A normal model is an assumption about a population distribution, not an automatic property of data.
Choose a defensible sample
A sampling plan connects the available observations to a target population. Random selection helps reduce selection bias; it does not guarantee that any one sample perfectly resembles the population.
Design an experiment
An experiment deliberately assigns treatments. An observational study records conditions without assigning them. Random assignment helps balance confounding variables across treatment groups in the long run.
Condition on the right group
Probabilities describe long-run relative frequencies under a specified chance process. A condition changes which outcomes remain possible and therefore changes the denominator.
Model repeated chance outcomes
A random variable assigns a numerical outcome to a chance process. Its expected value is a probability-weighted mean over many repetitions, not a promised outcome of one trial.
Separate data from statistics
A sampling distribution is the distribution of a statistic across repeated samples of the same size. It differs from the distribution of individual observations in a single sample.
Build and interpret intervals
A confidence interval combines an estimate with uncertainty from sampling. A common structure is estimate plus or minus a critical multiplier times standard error. The interpretation must name the population parameter and context.
Reason from a null model
A significance test asks whether sample results are compatible with a specified null model. The alternative hypothesis determines what counts as more extreme. A test decision is not proof of a hypothesis.
Compare groups carefully
Comparing groups requires a clear parameter: a difference in proportions, a difference in means, or an association between categorical variables. The design determines which observations can be treated as independent.
Read a fitted relationship
A fitted regression line summarizes a linear association and predicts a response from an explanatory variable. The slope describes the predicted response change for a one-unit increase in the explanatory variable.
Read a distribution, not just its average
Read frequencies from intervals, compare distributions fairly, and distinguish a descriptive pattern from a claim about the population. A histogram groups numerical measurements into adjacent intervals. A bar chart instead separates categories. The distinction matters because the horizontal distance has a numerical meaning only in the histogram.
Repair the data-collection design
Choose a sampling or assignment procedure that answers the intended question. A plan can be random in one respect and biased in another. Before choosing a procedure, state the target population, available sampling frame, response measurement and possible predictors of the response.
Compare conditional distributions
Use conditional distributions to investigate association between categorical variables. One variable defines the groups being compared, and the other describes the outcome within each group. A grand-total percentage may be useful, but it does not by itself compare the groups.
Build and audit a simulation
Build a simulation that reproduces the stated chance process, and distinguish a simulated estimate from an exact theoretical probability. Define one trial, one repetition of the full experiment, the statistic recorded, and the event whose frequency will be counted.
Check a binomial model before calculating
Use a binomial model only after identifying a fixed number of trials, two outcomes per trial, independence and a common success probability. The random variable counts successes. A total mass or a waiting time until the first success is a different kind of variable.
Move between percentiles and measurements
Move between normal-model areas and original measurements without confusing a z score with a probability. A percentile describes cumulative area below a cutoff. The cutoff has measurement units; a z score expresses its distance from the mean in standard-deviation units.
Audit the assumptions behind an interval or test
Check the assumptions of the specific inference method rather than reciting a generic list. Random collection, approximate independence and a suitable sampling-distribution shape address different issues. Passing one condition does not imply that the other conditions hold.
Write a defensible population claim
Turn a test result into a statement about the population question. Write the parameter and the direction of the alternative before collecting data, state the significance level in advance, and keep the distinction between evidence and certainty visible in the conclusion.
Choose a t procedure from the study design
Choose a t procedure using the observational unit and the relationship between measurements. The target can be one population mean, a mean within-person difference, or a difference between independent population means. The unknown population SD is estimated from the appropriate sample.
Compare populations with the right uncertainty
Compare two population proportions while preserving the group labels and collection design. The parameter pA−pB has a direction. Positive values mean A has the higher population proportion; reversing the labels reverses both the estimate and the interval endpoints.
Test a categorical association as an investigation
Plan a chi-square investigation from how the observations were collected. An independence test usually records two categorical variables from one random sample. A homogeneity test compares one categorical response across separately sampled populations or randomized treatment groups.
Decide what a fitted line can support
Assess a fitted line with its original scatterplot, residuals and prediction domain. A compact equation can hide systematic error. Treat coefficients as descriptions of a fitted relation first; causal claims require an appropriate design beyond a strong numerical association.
Foundation pilot, not complete AP syllabus coverage or an official exam simulation.
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
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Scope: 2026–27 statistics practice: expanded partial coverage