Statistics foundations: data, design and sampling variability
Eighteen lessons build statistical investigations, descriptive analysis, probability models and sampling-distribution readiness. This is partial preparation for the Fall 2026 five-unit AP Statistics framework, not complete AP coverage. Confidence intervals, hypothesis-testing procedures and cumulative inference assessment need the separate inference sequence. Original datasets here are synthetic.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Statistics: design, variation and model selection
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Ask a question that expects variation
Lesson details and objectives
Define units, variables, populations and operational measurements before summarizing.
- Formulate a statistical question.
- Classify variables by their meaning rather than their storage format.
- Distinguish observations, missingness and replication.
6 practice tasks.
Choose who can enter the sample
Lesson details and objectives
Separate the target population, frame, selection and response processes.
- Describe simple random, stratified and cluster samples.
- Identify undercoverage and nonresponse.
- Separate sampling variability from systematic bias.
6 practice tasks.
Use design to support a causal claim
Lesson details and objectives
Plan controls, random assignment, replication, blocking and paired comparisons.
- Separate random selection from assignment.
- Construct blocked and paired designs.
- Identify confounding and appropriate inference scope.
6 practice tasks.
Change the denominator deliberately
Lesson details and objectives
Build joint, marginal and conditional proportions from a two-way table.
- Reconstruct totals and conditional proportions.
- Compare proportions rather than raw counts.
- Explain association and aggregation effects.
6 practice tasks.
Choose a display that preserves the question
Lesson details and objectives
Read bins, density, quartiles and outlier conventions honestly.
- Construct frequency and density histograms.
- Use an explicit quartile convention.
- Identify information lost in a boxplot.
6 practice tasks.
Summarize variation without hiding extremes
Lesson details and objectives
Calculate mean, median, sample variance and robust alternatives.
- Distinguish population and sample variance.
- Explain outlier sensitivity.
- Compare distributions in context with units.
6 practice tasks.
Locate a value within a model
Lesson details and objectives
Use z scores, normal areas and quantiles with explicit assumptions.
- Standardize and reverse standardization.
- Use given normal areas without confusing density and probability.
- Check whether a normal model is plausible.
6 practice tasks.
Describe a relationship before fitting it
Lesson details and objectives
Separate direction, form, strength and unusual observations.
- Read paired quantitative data.
- Interpret correlation as linear association.
- Identify unit invariance, nonlinearity and confounding.
6 practice tasks.
Fit a line and inspect its failures
Lesson details and objectives
Connect a least-squares model to residuals, influence and prediction limits.
- Compute slope and intercept from deviation sums.
- Interpret residuals with units.
- Diagnose patterns without turning association into causation.
6 practice tasks.
Change units without changing the question accidentally
Lesson details and objectives
Track centers, spreads and multiplicative patterns under transformations.
- Transform mean and SD correctly.
- Use logarithms for positive multiplicative data.
- Distinguish transformed averages from averages on the original scale.
6 practice tasks.
Build a coherent probability model
Lesson details and objectives
Use sample spaces, complements and unions without double counting.
- State equally likely assumptions.
- Apply inclusion–exclusion.
- Distinguish disjoint events from independent ones.
6 practice tasks.
Restrict the reference set before multiplying
Lesson details and objectives
Use conditional probabilities and independence with positive denominators.
- Compute conditional probability.
- Distinguish sampling with and without replacement.
- Test independence using a stated model.
6 practice tasks.
Update a probability without losing the base rate
Lesson details and objectives
Build natural-frequency trees and interpret reverse conditionals.
- Partition an outcome across sources.
- Apply Bayes with a nonzero evidence probability.
- Explain sensitivity, false positives and prior dependence.
6 practice tasks.
Average outcomes with probability weights
Lesson details and objectives
Construct a distribution and distinguish expectation from a guaranteed result.
- Validate probability masses.
- Compute expectation and variance.
- Interpret transformations and decision limitations.
6 practice tasks.
Check trials before counting successes
Lesson details and objectives
Use binomial counts only when the trial mechanism supports them.
- Check fixed n, binary trials, independence and constant p.
- Calculate binomial masses and complements.
- Distinguish model failure from ordinary variation.
6 practice tasks.
Distinguish the data from the statistic
Lesson details and objectives
Construct a small exact sampling distribution and identify bias and variability.
- Separate population, sample and sampling distributions.
- Enumerate repeated independent samples.
- Relate sample size to standard error under stated assumptions.
6 practice tasks.
Separate concentration from normal approximation
Lesson details and objectives
Explain what repeated sampling does and does not guarantee.
- State finite-moment independent-sampling versions of LLN and CLT.
- Distinguish observations from averages.
- Evaluate approximation conditions rather than invoke a universal threshold.
6 practice tasks.
Decide what the evidence can support next
Lesson details and objectives
Audit design, independence, model conditions and uncertainty before selecting inference.
- Distinguish parameter, statistic and standard error.
- Check proportion normal-approximation and sampling-fraction conditions.
- Separate interval coverage, tests and practical importance.
6 practice tasks.