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Also known as: CLT
A theorem stating that sums or means of many independent variables converge toward a normal distribution under mild conditions.
DESCRIPTION
The central limit theorem explains the ubiquity of the normal distribution in aggregated measurements. It states that properly normalized sums or averages of i.i.d. variables with finite variance approach normality as sample size grows. Variants extend to non-identical distributions under Lindeberg or Lyapunov conditions. CLT justifies normal-based inference, confidence intervals, and hypothesis tests. Finite-sample accuracy is bounded by results like Berry–Esseen.
ALTERNATIVE DEFINITIONS
NIST Engineering Statistics Handbook
The central limit theorem states that the distribution of the sample mean will be approximately normal for large sample sizes regardless of the distribution of the population.
Focuses on sample mean approximation regardless of underlying distribution.
StatLect
The central limit theorem states that the sum (or average) of a large number of independent and identically distributed random variables with finite variance is approximately normally distributed.
Mentions i.i.d. and finite variance conditions explicitly.
Encyclopedia of Mathematics
One of the basic results of probability theory, stating that the sum of a large number of independent random variables has an approximately normal distribution.
Concise encyclopedia-level statement of the result.
RELATED TERMS
Normal Distribution Sample Mean Lindeberg–Feller Theorem Berry–Esseen Theorem t-Distribution Bootstrap
COMPETING TERMS
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist Data Analyst AI Engineer
EXTERNAL SOURCES
- Wikipedia - Central limit theorem
- NIST - CLT explanation and examples
- StatLect - Detailed CLT derivations
- MIT OpenCourseWare - Lecture on CLT with notes
TOP QUESTIONS
Why is the Central Limit Theorem important?
It justifies normal approximations for sums and means, enabling standard inferential methods even when data are not normally distributed.
Category: Relevancy
What are the assumptions for the CLT?
Typically independent, identically distributed variables with finite variance; generalized versions relax i.i.d. via conditions like Lindeberg’s.
Category: Requirements
How large must n be for the CLT to hold?
It depends on the distribution’s skewness and tails; rules of thumb like n≥30 are common but not universal.
Category: Misunderstanding
Does the CLT apply to medians?
Variants of asymptotic normality apply to many estimators under regularity, but the classical CLT targets sums and means.
Category: Concept
What is the Berry–Esseen bound?
It quantifies the rate at which distributions converge to normal under the CLT, depending on third moments.
Category: Advanced
How does CLT relate to the Law of Large Numbers?
LLN ensures convergence of averages to the mean; CLT describes the distribution around that mean.
Category: Differences
Can heavy-tailed data violate the CLT?
If variance is infinite, the classical CLT fails and stable laws can arise instead.
Category: Risks
How does CLT support confidence intervals?
Approximate normality of sample means enables normal or t-based intervals for population means.
Category: Application
Is bootstrapping a replacement for CLT?
Bootstrapping approximates sampling distributions empirically; it complements rather than replaces CLT-based theory.
Category: Opinion