Hassan Ijaz
Ai, Web & Design
Law of large numbers and central limit theorem
Animated simulation showing sample means converging to population mean, with adjustable sample sizes and various starting distributions
Concept Overview
The Law of Large Numbers is a fundamental theorem that explains why sample averages converge to population means as sample sizes increase. It provides the theoretical foundation for statistical inference.
Two Forms of the Law
Weak Law of Large Numbers
For any ε > 0, P(|X̄ₙ - μ| > ε) → 0 as n → ∞
- Sample mean converges in probability to population mean
- Probability of large deviations approaches zero
- Applies to any distribution with finite variance
Strong Law of Large Numbers
P(X̄ₙ → μ as n → ∞) = 1
- Sample mean converges almost surely to population mean
- Stronger guarantee than weak law
- Sample path will eventually stay close to μ
Central Limit Theorem
Related but different: CLT describes the distribution shape, not just convergence
√n(X̄ₙ - μ) → N(0, σ²) in distribution
- Sample means are approximately normal for large n
- Works regardless of original distribution shape
- Foundation for confidence intervals and hypothesis tests
Rate of Convergence
Standard error decreases as 1/√n:
- SE(X̄) = σ/√n
- To halve standard error, need 4× sample size
- Diminishing returns to larger samples
- Trade-off between precision and cost
Practical Applications
Quality Control
Monitor process averages over time
Polling
Estimate population preferences from samples
Monte Carlo Methods
Approximate integrals using sample averages
Insurance
Predict claims based on large portfolios
Important: The law doesn't guarantee that every sample will be close to the mean - it says that the probability of being far away becomes smaller as n increases.
Watch the animated simulation below as sample means converge to the population mean. Try different starting distributions and sample sizes to see the law in action.
Interactive Visualization
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Animated simulation showing sample means converging to population mean, with adjustable sample sizes and various starting distributions