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Central Limit Theorem

Repeatedly sample from a uniform population to show normal approximation and σ/√n.

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X̄ₙ ~ N(μ, σ²/n) approx.
each draw takes n values
compute m sample means

📖 Tutorial | Central Limit Theorem

1. Definition
Central Limit Theorem (CLT): i.i.d. sampling from a population with mean μ, variance σ²; for large n the sample mean X̄ₙ is approx. N(μ, σ²/n), whatever the population shape.
2. Symbols
SymbolMeaning
X̄ₙsample mean of size n
μpopulation mean (here 0.5)
σpopulation sd (here 1/√12)
σ/√nstandard error
3. How it works
  • Draw m samples of size n from uniform U(0,1);
  • Compute mean and sd of these m sample means;
  • Their mean should be near 0.5;
  • Their sd should be near σ/√n;
  • The shape is bell-shaped.
4. Steps
  1. Enter sample size n, e.g. 30;
  2. Enter number of draws m, e.g. 10000;
  3. Click Calculate and compare observed vs theory;
  4. Increase n to see the SE shrink.
5. Example
Example: n=30, uniform U(0,1) mean 0.5, σ=1/√12≈0.2887. Theoretical SE σ/√30≈0.0527; with m=10000 the observed sd is about 0.05.
6. Pitfalls
Small n (e.g. 5) gives a poor normal approximation;
m affects stability but not the theoretical σ/√n;
A fixed seed keeps results reproducible.

❓ FAQ | Central Limit Theorem

Why is the mean approximately normal?
Many independent small fluctuations sum to a normal average by the CLT.
What is standard error?
SE=σ/√n, the sd of the sample mean, measuring how estimation error shrinks with n.
How large must n be?
Roughly n≥30; skewed populations need larger n.
Which population is used?
Uniform U(0,1), mean 0.5 and variance 1/12.
Relation to the LLN?
LLN says the mean converges to μ; CLT describes the normal shape of the fluctuation.