Estimation and uncertainty

From a population limit to a justified standard error

An estimator is a random rule applied to a sample. Its probability limit names the quantity it approaches as information grows; its sampling distribution describes variation across repeated studies. A small standard error does not establish that the limit equals the scientific target.

For a completely observed population mean, the estimation error is exactly an average of centered observations. Linearization extends this idea to sample equations and smooth transformations. A negligible remainder, a suitable central limit theorem, and consistent estimation of the contribution variance then support uncertainty statements. The worked arguments connect each theorem to the step it justifies.

What you’ll learn

• Connect sample and population: use laws of large numbers and a population equation to understand what an estimator approaches.
• Justify and interpret uncertainty: derive a first-order contribution, control the remainder, estimate its variance, and interpret confidence intervals and tests.

The core completion targets begin with LS-01–LS-04: interpret convergence, expand expectation notation, distinguish SD from SE, and combine the working theorems. SP-01 then develops the smooth-root argument, and INF-01 checks uncertainty interpretation. Cluster inference and bootstrap apply the same reasoning to particular designs and procedures; their checkpoints are completion targets when you work through those extensions. The bridge and full checkpoint catalog are references, rather than prerequisites at the start.