--- name: math-thumb-statistics description: "Apply Chapter 13 (Statistics: From Data to Defensible Decisions) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to plan precision and report uncertainty using the actual independent sampling units." --- # Statistics: From Data to Defensible Decisions Use this chapter to help the reader make a checked mathematical decision. All 49 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Estimand and target population; sampling or assignment design; independent units; variation; precision target; multiplicity and stopping plan. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule 1. **Name the target.** Is it a finite-population total, a future-population mean, a treatment contrast, a proportion, a correlation, a predictive loss, or a model coefficient? Do not calculate until units and target population are explicit. 2. **Map the observation structure.** Mark clusters, repeated measurements, time order, strata, weights, blocks, pairs, censoring, and finite sampling fractions. Decide what the independent units really are. 3. **Choose the scale and summary.** Use arithmetic methods for additive variation, logarithms or geometric means for multiplicative variation, and resistant summaries when tails can dominate. 4. **Plan precision.** Select a meaningful margin or effect, estimate nuisance variation conservatively, apply the correct allocation and design effect, add attrition, and round upward. 5. **Choose inference from the design.** Use \(t\) when scale is estimated, Wilson or plus-four for ordinary proportions, exact or simulated table methods when sparse, and sequential boundaries for repeated looks. 6. **Declare the claim family.** Decide whether the goal is one protected conclusion, familywise control, or discovery screening. Set equivalence margins and multiplicity rules before outcomes are inspected. 7. **Fit and compare models.** Use AIC, BIC, or cross-validation only among scientifically plausible candidates fitted on comparable data. Preserve preprocessing inside resampling. 8. **Stress-test the answer.** Examine VIF, leverage, Cook’s distance, events per parameter, bootstrap convergence, tail rank, and sensitivity to flagged observations or design assumptions. 9. **Report the decision range.** Give the estimate, interval, assumptions, denominator, design, adjustment family, and the values within the uncertainty range that would reverse the decision. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Plan for the independent units and claim family. Screening thresholds do not automatically justify deletion, equivalence, model validity, or a universal sample-size guarantee. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 13.1.5, 13.1.8, 13.2.9 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C13-D01: Count independent information rather than readings**: rule 13.2.9. - **C13-D02: Put an upper bound on zero observed events**: rule 13.1.8. - **C13-D03: Compare interval behavior near zero and one**: rule 13.1.9. - **C13-D04: Count the cost of peeking at p-values**: rule 13.3.3. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.