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5.3 The Central Limit Theorem

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The Central Limit Theorem

The Central Limit Theorem (CLT)

Formal statement: If X1,X2,…,Xn{X_1, X_2, \ldots, X_n} are i.i.d. with mean μ{\mu} and standard deviation σ{\sigma}, then for large n{n}:
Xˉ≈N ⁣(μ, σn)\bar{X} \approx N\!\left(\mu,\ \frac{\sigma}{\sqrt{n}}\right)

Conditions for the CLT to apply:

  • Sample size n≥30{n \geq 30} (rough guideline for non-normal populations)
  • Individual observations must be independent (check: 10% condition)

Why is the CLT so important?

  • It allows inference even when the population is not normal
  • It explains why the normal distribution appears so often in practice
  • It is the theoretical foundation for confidence intervals and hypothesis tests in Units 6-9

In this unit

8 lessons

  1. 5.1Introducing Statistics: Why Is My Sample Not Like Yours?Preview lesson
  2. 5.2The Normal Distribution, RevisitedPreview lesson
  3. 5.3The Central Limit TheoremThis lesson
  4. 5.4Biased and Unbiased Point EstimatesPreview lesson
  5. 5.5Sampling Distributions for Sample ProportionsPreview lesson
  6. 5.6Sampling Distributions for Differences in Sample ProportionsPreview lesson
  7. 5.7Sampling Distributions for Sample MeansPreview lesson
  8. 5.8Sampling Distributions for Differences in Sample MeansPreview lesson

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