Chebyshev's Inequality

Probability · Axiom Academy

Powerful variance-based bound that quantifies how tightly values cluster around the mean 1. Statement of Chebyshev's Inequality For any random variable X with finite mean μ and variance σ², and for any k > 0: Alternative form: P(|X - μ| ≥ ε) ≤ σ²/ε². As the allowed deviation ε increases, the bound decreases quadratically. 2. Proof via Markov's Inequality Chebyshev's inequality is derived by applying Markov's inequality to the non-negative random variable (X - μ)²: The key insight: |X - μ| ≥ kσ if and only if (X - μ)² ≥ k²σ². Apply Markov to (X - μ)² with threshold k²σ², noting E[(X - μ)²] = σ². Chebyshev's inequality gives concrete guarantees about concentration around the mean: These bounds hold for any distribution with finite variance, making Chebyshev universally applicable but sometimes conservative. A manufacturing process produces parts with mean weight μ = 100g and standard deviation σ = 5g. What proportion of parts weigh between 85g and 115g? The range [85, 115] is μ ± 3σ (k = 3). By Chebyshev, at most 1/9 ≈ 11% of parts fall outside this range, so at least 89% fall within it. The actual percentage depends on the distribution shape. 5. Connection to Law of Large Numbers Chebyshev's inequality provides the key step in proving the weak law of large numbers. For sample mean X̄_n of n i.i.d. variables: Since Var(X̄_n) = σ²/n, Chebyshev gives P(|X̄_n - μ| ≥ ε) ≤ σ²/(nε²) → 0 as n → ∞. The sample mean converges in probability to the true mean.

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