Three Sigma Without a Distribution

Question

With no distributional assumptions at all, only that the mean mu and variance sigma^2 exist, what is the largest possible value of P(|X - mu| ge 3sigma)? Compare it with the normal answer, and exhibit a distribution that attains the bound.

Accepted Answer

Chebyshev via Markov. (X-mu)^2 ge 0 with mean sigma^2, so Markov gives Pbig((X-mu)^2 ge 9sigma^2big) le fracmathbbE[(X-mu)^2]9sigma^2 = frac19 approx mathbf11.1%. Versus the normal: P(|Z| ge 3) approx 0.27%, a factor of roughly 41 smaller. The gap between 0.27% and 11.1% is the entire playground of fat-tailed distributions; daily returns live somewhere inside it, far from the normal end. The bound is tight. Put mass frac118 at mu + 3sigma, mass frac118 at mu - 3sigma, and mass frac89 at mu. Mean is mu by symmetry; variance is 9sigma^2 cdot frac19 = sigma^2 ✓; and P(|X-mu| ge 3sigma) = frac19 exactly. No distribution-free improvement is possible. The trap: quoting "a 3σ day happens once every 370 days" as if it were a law of probability. That frequency is a normality statement. The assumption-free statement is only "at most once every 9 days on average," and real markets sit in between, which is why risk teams treat 3σ moves as monthly-ish events, not yearly ones. Desk relevance: Chebyshev is the honest floor under every risk conversation, it is what you can promise a risk committee without a model, and the distance between it and your model's tail is exactly how much you are trusti

Three Sigma Without a Distribution

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