Statistical Inference

An estimand is the quantity an analysis intends to learn, such as a population mean, difference or model parameter. An estimator maps a sample to an estimate, and its sampling behavior determines uncertainty and possible bias. Confidence intervals, hypothesis tests and resampling methods offer different ways to summarize that uncertainty. Bayesian inference instead conditions on a model and prior to describe posterior uncertainty. Both require a defensible connection between observations and the target question. Inference is distinct from merely computing a sample statistic: the additional claim concerns unobserved units, repeated sampling or unknown quantities, and must be justified by the design and assumptions.

Specify the population and estimand, then inspect how units were selected and whether observations are independent or clustered. Choose an estimator and uncertainty calculation compatible with skew, missingness and dependence. Use diagnostic plots and, where useful, simulation to test the analysis under plausible data conditions. Report an effect size and interval with a clear interpretation, rather than only a p-value. The result should expose the route from the observed sample to the broader statement, including sensitivity to assumptions that the data cannot directly verify.

An interval can be narrow around a biased estimate if the sample is unrepresentative or the model is misspecified. Independence assumptions, unmodeled clustering and selective missingness can invalidate ordinary error calculations. Bootstrapping does not automatically fix the sampling design. A frequentist confidence interval is not a posterior probability interval for the realized parameter. Inference also does not establish causation without appropriate identification. State the assumptions and avoid extrapolating beyond populations and conditions the study can support.