Any random variable X with finite variance σ² satisfies P(|X − E[X]| ≥ kσ) ≤ 1/k². Chebyshev 1867. Distribution-free; loose for light-tailed X (e.g. bound 1/9 vs. exact 0.0027 for normal at k=3) but sharp at two-point X ∈ {−kσ, kσ}. …
Any random variable X with finite variance σ² satisfies P(|X − E[X]| ≥ kσ) ≤ 1/k². Chebyshev 1867. Distribution-free; loose for light-tailed X (e.g. bound 1/9 vs. exact 0.0027 for normal at k=3) but sharp at two-point X ∈ {−kσ, kσ}. …