Lesson 14 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 14 of 23 · 14 min

Two means: independent vs paired samples

Before comparing two means, decide whether the samples are independent (pooled t-test on the difference in means) or related (paired comparisons test on the mean of the differences).

In short

  • Independent samples from normal populations with unknown but equal variances: pooled t-test with n1+n2−2n_1 + n_2 - 2 df.
  • The pooled variance sp2s_p^2 is a df-weighted average of the two sample variances.
  • Dependent (paired) samples, such as two assets over the same dates or the same firms before and after: compute each pair's difference and run a paired comparisons test with n−1n - 1 df.
  • On related data the paired test is more powerful, because differencing strips out variation the samples share.
  • Usual null: no difference (μ1−μ2=0\mu_1 - \mu_2 = 0 or μd=0\mu_d = 0). One-sided versions work just as for a single mean.

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