Chebyshev's inequality

Chebyshev's inequality limits the probability of deviation from the mean

Chebyshev's inequality provides an upper bound on the probability that a random variable deviates from its mean by more than k standard deviations. This inequality applies to any probability distribution with a defined mean and variance, making it a versatile tool in probability theory. It helps establish the weak law of large numbers by demonstrating that the probability of significant deviations decreases as the sample size increases.

Example

Suppose we have a random variable X with mean μ and variance σ². Chebyshev's inequality states that the probability P(|X - μ| ≥ kσ) ≤ 1/k². For instance, if k = 2, the probability that X deviates from its mean by at least 2 standard deviations is at most 1/4 or 25%.

Understanding Chebyshev's inequality helps in assessing the reliability of statistical estimates and recognizing the limitations of data processing.

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