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Chebyshev's Inequality

Formula

\[ P(|X-\mu| \ge k\sigma) \le \frac{1}{k^2},\quad k>0 \]

Parameters

  • \(\mu=\mathbb{E}[X]\)
  • \(\sigma=\sqrt{\operatorname{Var}(X)}\)

What it means

Bounds the probability of large deviations using only mean and variance.

What it's used for

  • Bounding tail probabilities with only mean/variance.
  • Concentration guarantees without distribution assumptions.

Key properties

  • Distribution-free
  • Often loose for light-tailed distributions

Common gotchas

  • Applies to any distribution with finite variance.
  • Use sharper bounds when additional assumptions are known.

Example

If \(\sigma=2\), then \(P(|X-\mu|\ge 4)\le (2^2/4^2)=1/4\).

How to Compute (Pseudocode)

Input: assumptions/quantities required by the theorem or inequality (for example means, variances, sample size)
Output: bound, approximation, or theorem-based diagnostic

verify the theorem/inequality assumptions (at least approximately/in modeling terms)
compute the bound or approximation using the card formula
return the resulting bound/approximation and note its conditions

Complexity

  • Time: Usually \(O(1)\) once the required summary quantities are available
  • Space: \(O(1)\)
  • Assumptions: This is a formula-application workflow; estimating required moments/parameters from data can dominate cost (often \(O(n)\))

See also