Chebyshev inequalities

E451512

concentration inequality probabilistic inequality

Chebyshev inequalities are probabilistic bounds that limit how much a random variable’s values can deviate from its mean in terms of its variance.

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Observed surface forms (2)

Surface form	Occurrences
Chebyshev inequality	1
Chebyshev’s theorems	1

Statements (46)

Predicate	Object
instanceOf	concentration inequality ⓘ probabilistic inequality ⓘ
alsoKnownAs	Chebyshev–Cantelli inequalities NERFINISHED ⓘ Chebyshev’s inequality NERFINISHED ⓘ
appliesTo	both discrete and continuous random variables ⓘ random variables with finite variance ⓘ
assumes	finite second moment ⓘ
category	inequalities in probability theory ⓘ
doesNotAssume	independence of observations ⓘ symmetry of distribution ⓘ
doesNotRequire	specific distributional assumptions ⓘ
field	probability theory ⓘ statistics ⓘ
foundationFor	other concentration inequalities ⓘ
generalizes	empirical rule to arbitrary distributions with finite variance ⓘ
guarantees	at least 1−1/k² of probability mass lies within k standard deviations of mean ⓘ
hasForm	P(\|X−μ\| ≥ kσ) ≤ 1/k² for k>0 ⓘ
hasOneSidedForm	P(X−μ ≥ kσ) ≤ 1/(1+k²) in Cantelli form ⓘ
hasProperty	applies for all k>0 ⓘ tight for some distributions ⓘ
implies	at least 75% of values lie within 2 standard deviations of mean ⓘ at least 89% of values lie within 3 standard deviations of mean ⓘ at least 96% of values lie within 5 standard deviations of mean ⓘ probability of large deviation is small if variance is small ⓘ
is	distribution-agnostic ⓘ often loose compared to distribution-specific bounds ⓘ
isSpecialCaseOf	Markov inequality NERFINISHED ⓘ
namedAfter	Pafnuty Chebyshev NERFINISHED ⓘ
provides	upper bound on tail probabilities ⓘ
relatedTo	Chernoff bounds NERFINISHED ⓘ Hoeffding inequality NERFINISHED ⓘ Markov inequality NERFINISHED ⓘ
relates	variance to deviation from mean ⓘ
type	moment inequality ⓘ
usedFor	bounding error probabilities in algorithms ⓘ constructing confidence bounds ⓘ distribution-free bounds ⓘ proving weak law of large numbers ⓘ robust risk assessment ⓘ sample size estimation ⓘ
usedIn	actuarial science ⓘ finance ⓘ machine learning ⓘ quality control ⓘ
uses	mean of a random variable ⓘ variance of a random variable ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Inequalities → containsTopic → Chebyshev inequalities ⓘ

Pafnuty Chebyshev → notableWork → Chebyshev inequalities ⓘ

this entity surface form: Chebyshev inequality

Mertens’ theorems → relatedTo → Chebyshev inequalities ⓘ

this entity surface form: Chebyshev’s theorems