2.8 Summary

• A population has fixed but unknown parameters.
• An estimator is a rule; an estimate is the result of applying it to one sample.
• Since the truth is never observed, estimators are judged by their behaviour across repeated samples, not by any single estimate they produce.
• A sample speaks for a population only if it is drawn independently and from the population of interest. Neither assumption is decoration: break the first and the spread is far wider than any formula predicts, break the second and the estimates are centred on the wrong number while often looking more precise than the honest ones.
• Under random sampling the sample mean is unbiased: \(E(\bar{X}) = \mu\).
• Its standard error, \(\sigma/\sqrt{n}\), falls as \(1/\sqrt{n}\) — so larger samples give more precise estimates, at a rising price per unit of precision.
• Centred and increasingly precise together are what make the sample mean trustworthy, and together they are the Law of Large Numbers.

Everything in this unit has been about choosing a good estimator. The next unit asks how to use that estimator to make statements about an unknown population from a single observed sample.