Lesson 7 of 23 · All lessons in this module
  1. The central limit theorem
  2. The standard error of the sample meanVideo lesson
  3. Confidence intervals for the mean
  4. Value at risk: a one-sided confidence interval
  5. Populations, samples and sampling error
  6. Stratified and cluster sampling
  7. Good estimators, and the CLT across sampling designs
  8. Convenience and judgmental sampling, and choosing a method
  9. The four steps and stating the hypotheses
  10. Type I and Type II errors, significance and powerVideo lesson
  11. Critical values, p-values and the decision ruleVideo lesson
  12. p-values and p-hacking
  13. Testing a single mean
  14. Two means: independent vs paired samplesVideo lesson
  15. Testing variances: chi-square and FVideo lesson
  16. The t-test for a Pearson correlation
  17. Sample size, p-values and many correlations at once
  18. Parametric vs nonparametric tests
  19. The Spearman rank correlation testVideo lesson
  20. Contingency tables and the chi-square test of independenceVideo lesson
  21. Interpreting the result: standardized residuals and mosaics
  22. Choosing the test: four steps and a map of every test
  23. Integrated cases: testing whether a new strategy beats the old one

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 7 of 23 · 13 min

Good estimators, and the CLT across sampling designs

A good estimator is unbiased, efficient, consistent and, when the data are messy, robust; and how fast the sample mean becomes normal depends on the population's shape and on how the sample was drawn.

In short

  • Unbiased: the estimator's expected value equals the parameter. The sample mean is unbiased for μ\mu.
  • MVUE (minimum-variance unbiased estimator): unbiased with the smallest variance of all unbiased estimators. BLUE (best linear unbiased estimator): the most efficient among unbiased estimators that are linear combinations of the data.
  • Consistent: converges to the true value as n grows. Robust: stays reliable with outliers or non-normal data, for example a trimmed mean.
  • Sample size the CLT needs: about 4–5 for symmetric, single-peaked populations; 25–30 for skewed ones; more for extreme skew.
  • Stratified: the CLT works within each stratum and for the combined mean. Cluster: it depends on the size of the clusters, not the number of clusters. Non-probability samples: the CLT does not reliably apply.

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