Central Limit Theorem

Probability · Axiom Academy

Why the normal distribution appears everywhere in nature and statistics 1. Statement of the Central Limit Theorem Let X₁, X₂, ..., Xₙ be i.i.d. random variables with mean μ and finite variance σ². Define: Then the standardized sample mean converges in distribution to a standard normal: 2. From Any Distribution to Normal The remarkable aspect of the CLT is that it works regardless of the original distribution of Xᵢ. Whether the data is: Or any other distribution with finite mean and variance The sum or average of sufficiently many samples will always look approximately normal! Important: The original distribution doesn't need to be symmetric or bell-shaped. The magic happens through aggregation. 3. Conditions and Practical Interpretation Independence: The samples X₁, X₂, ... must be independent Identical distribution: All Xᵢ have the same distribution Finite variance: σ² < ∞ (the variance must exist and be finite) For symmetric distributions: n ≥ 30 is often sufficient For skewed distributions: n ≥ 50 or more may be needed For highly skewed data: n ≥ 100 recommended

This is the written version of the interactive lesson above. See the full Probability course.