Populations, Samples, and Sampling Distributions

Every time a risk manager estimates a portfolio's mean return from three years of daily data, or a quantitative researcher calibrates a model to a window of observed prices, they are using a sample — a finite draw from a much larger population of possible outcomes — to make inferences about that population. The central challenge is that the estimates themselves are random: a different sample would give different numbers. Understanding the probability distribution of an estimator, called its sampling distribution, is the engine of all classical inference: it says how far a typical estimate is from the truth, and by how much.

This note builds the foundation. It distinguishes populations from samples, defines summary statistics and the key measures — sample mean $\bar{X}$, sample variance $S^2$, standard error — and derives their exact sampling distributions under normality. The Central Limit Theorem refines the picture for large samples from any distribution. The chi-squared, $t$, and $F$ distributions that arise here are studied in depth in the next lesson.

Learning objectives. By the end of this note you should be able to: