Choose the question before the formula

Try

Three studies each compare two columns: a before-and-after time study, two independently sampled groups’ times, and two independent groups’ binary outcomes. Their tables look similar, but the inferential targets and dependence differ. Identify the observational unit and what a single recorded value means before choosing a method.

Predict

Predict why equal sample sizes do not distinguish paired from independent samples. Pairing comes from the design: matched people, repeated measures or a defined matching rule. Also predict why coding success as 1 and failure as 0 does not make every generic small-sample t method the best introductory choice for binary inference.

Experiment

Build a method-selection ledger.

Question	Response and design	Introductory method, if conditions hold
One population success rate	Binary independent observations	One-proportion interval or test
Difference in independent rates	Binary, separate groups	Two-proportion inference
Mean within-pair change	Quantitative matched differences	One-sample t on differences
Difference in independent means	Quantitative, separate groups	Welch two-mean inference
Association of two categories	One sample, two categorical variables	Chi-square independence

Add the parameter and sampling design beside each row. The table suggests a starting point; it is not a guarantee that every listed dataset satisfies the method’s assumptions.

Understand

A defensible analysis begins with the scientific question and target population or process. Decide whether the task is estimation, a prespecified hypothesis test or a decision requiring an effect threshold. Define the parameter in words and symbols. A difference needs an orientation; a change needs a timing and sign convention; a categorical outcome needs a consistent definition.

Next identify the unit of independent information. Repeated visits, clustered classrooms and matched pairs are not interchangeable with independent individuals. Count complete pairs when analyzing differences. If the design is complex, the simple procedures in this unit may not apply, even when a calculator accepts the summary values.

Check data type and structure. Quantitative measurements support mean comparisons when a mean is an appropriate target. Binary outcomes support proportions. Multi-category responses produce contingency tables. Categories encoded as integers do not acquire meaningful numerical distances; averaging transport codes is not a substitute for comparing distributions.

Then examine conditions for the chosen reference distribution: expected or observed counts for the relevant proportion procedure, expected cell counts for chi-square, and distributional shape and outliers for mean procedures. These checks supplement design validity. A large-count condition does not fix nonresponse bias; a symmetric histogram does not establish independent selection.

Missingness and selective exclusions deserve explicit attention. Analyzing only complete pairs can bias results if dropout relates to adverse outcomes. A record omitted for a documented measurement error is different from one omitted because it weakens significance. Preserve a transparent inclusion rule and consider sensitivity to plausible missingness mechanisms.

Finally calculate, interpret and evaluate limitations. A p-value is not a causal design, a confidence interval is not an individual guarantee, and a small standard error is not total uncertainty. Where the available method is inappropriate, a clear explanation of the mismatch is a stronger result than an unjustified numerical answer.

Apply

For each practice scenario, write a short plan containing target, method, analysis unit, conditions and permissible conclusion. Include at least one reason you might decline to calculate with the available information. For a workplace report, separate descriptive findings from population generalization and causal claims rather than letting a method name imply all three.

Continue

The next two lessons introduce resampling ideas. They can provide useful reference distributions, but must preserve the design and cannot manufacture information absent from the original data.