The Delta Method
How to get the variance of a function of an estimator, when the function itself has no simple variance rule
Opens with a precise primer on parameters, estimators, and the expected value and variance of an estimator, then derives the Delta Method from a first-order Taylor expansion: why Var(g(theta-hat)) is approximately [g'(theta)]^2 times Var(theta-hat), stated formally as an asymptotic-normality theorem, plus the brief multivariate/gradient version. Worked example transforms a binomial proportion into a log-odds -- the exact quantity behind every logistic-regression Wald SE -- and a 200,000-rep simulation shows skipping the derivative understates the SE by nearly 10x, while a naive confidence interval built directly on the proportion has both the wrong coverage and, a quarter of the time, an impossible negative endpoint. Closes with an honest look at how approximation quality depends on sample size, plus when a purpose-built alternative (e.g. Wilson score intervals) beats the general-purpose Delta Method.