# Chebyshev's inequality

**Chebyshev's inequality**, also called the **Bienaymé–Chebyshev inequality**, is a theorem of probability theory that bounds the probability that a random variable deviates from its expected value in terms of its variance. For any random variable with finite mean and finite variance and any ε > 0, the probability that the variable lies at least ε away from its mean is at most the variance divided by ε².<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup> Its value lies in its universality: it requires no assumption about the shape of the distribution, only finite mean and variance.<sup>[2](https://ai.stanford.edu/~gwthomas/notes/concentration.html)</sup>

| Key fact | Detail |
|---|---|
| Statement | For a random variable X with finite mean E(X) and variance Var(X), P(|X − E(X)| ≥ ε) ≤ Var(X)/ε² for any ε > 0<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup> |
| Standard-deviation form | The probability of falling more than k standard deviations from the mean, in either direction, is at most 1/k²<sup>[2](https://ai.stanford.edu/~gwthomas/notes/concentration.html)</sup> |
| Discovery | Found independently by I. Bienaymé (1853) and P. L. Chebyshev (1866 by one source, 1867 by others)<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup><sup> • </sup><sup>[3](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=820841)</sup> |
| Proof | A direct application of Markov's inequality to the nonnegative random variable (X − E[X])²<sup>[2](https://ai.stanford.edu/~gwthomas/notes/concentration.html)</sup> |
| Principal use | Proving the weak law of large numbers: the sample mean of n independent variables with variance σ² deviates from μ by more than t with probability at most σ²/(nt²)<sup>[4](https://faculty.washington.edu/yenchic/20A_stat512/Lec3_Expectation.html)</sup> |
| Sharpness | The bound cannot be improved: a symmetric three-point distribution attains equality for every variance and every k ≥ σ<sup>[5](https://physics2u.com/maths/vault/chebyshev-inequality.html)</sup> |
| Requirements | Needs only the mean and variance; unlike Chernoff or Hoeffding bounds, it does not require independence or bounded variables<sup>[6](https://courses.cs.washington.edu/courses/cse493s/25au/concentration-inequalities.pdf)</sup> |

## Statement

Let X be a random variable with finite mathematical expectation E X and variance D X. Chebyshev's inequality states that for any ε > 0 the probability of the event {|X − E X| ≥ ε} does not exceed D X/ε². Equivalently, writing the deviation in units of the standard deviation,

\[ \mathsf{P}\{|X - \mathsf{E}X| \geq t\sqrt{\mathsf{D}X}\} \leq \frac{1}{t^{2}}. \]

<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup> In the notation of mean μ and variance σ², the same result reads P(|X − μ| ≥ ε) ≤ σ²/ε².<sup>[4](https://faculty.washington.edu/yenchic/20A_stat512/Lec3_Expectation.html)</sup>

The standard-deviation form gives the interpretation used in elementary statistics: at least 1 − 1/k² of a distribution lies within k standard deviations of its mean, whatever its shape. No distribution can put more than a quarter of its probability two or more standard deviations from its mean, or more than a ninth beyond three, with no symmetry assumed and no bell curve required.<sup>[7](https://en.libre.university/lesson/probability/variance-and-spread)</sup>

## Proof

The standard proof applies [Markov's inequality](https://www.edgepedia.org/markovs-inequality), which says that a nonnegative random variable Z satisfies P(Z ≥ a) ≤ E(Z)/a, to the squared deviation:<sup>[2](https://ai.stanford.edu/~gwthomas/notes/concentration.html)</sup>

\[ \mathsf{P}(|X - \mathsf{E}[X]| \geq \epsilon) = \mathsf{P}\bigl((X - \mathsf{E}[X])^{2} \geq \epsilon^{2}\bigr) \leq \frac{\mathsf{E}\bigl[(X - \mathsf{E}[X])^{2}\bigr]}{\epsilon^{2}} = \frac{\operatorname{Var}(X)}{\epsilon^{2}}. \]

The first equality holds because the two events are identical, and the last step uses the definition of variance as E[(X − E[X])²].<sup>[2](https://ai.stanford.edu/~gwthomas/notes/concentration.html)</sup><sup> • </sup><sup>[8](https://cs.uwaterloo.ca/~r5olivei/courses/2025-spring-cs466/lecture-notes/lecture3/)</sup> The same argument can be run directly from the definition of expectation: on the set where |X − μ| ≥ ε, the squared deviation (X − μ)² is at least ε², so the variance, which averages (X − μ)² over the whole distribution, is at least ε² times the probability of that set; dividing by ε² gives the bound.<sup>[7](https://en.libre.university/lesson/probability/variance-and-spread)</sup>

## History

The inequality was discovered independently by the French mathematician I. J. Bienaymé in 1853 and by P. L. Chebyshev. The Encyclopedia of Mathematics dates Chebyshev's version to 1866,<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup> while a NIST account and the arXiv history of the law of large numbers date it to 1867.<sup>[3](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=820841)</sup><sup> • </sup><sup>[9](https://arxiv.org/abs/2208.08813)</sup> Bienaymé's paper of 1853 was reprinted in 1867 in Liouville's journal, immediately preceding the French version of Chebyshev's paper, and in a paper published in 1874 Chebyshev acknowledged Bienaymé's priority.<sup>[10](https://encyclopediaofmath.org/wiki/Bienaym%C3%A9%2C_Iren%C3%A9e-Jules)</sup>

Modern literature usually calls it Chebyshev's inequality, possibly because Chebyshev's name is attached to an application of it in the proof of the law of large numbers.<sup>[1](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)</sup> In the history of that law, the Bienaymé–Chebyshev inequality marks the meeting point of the French and Russian research directions.<sup>[11](https://arxiv.org/abs/1309.6488)</sup>

## Role in the law of large numbers

The inequality's most consequential use is the weak law of large numbers. In his 1867 work *On average quantities*, Chebyshev established that for independent random variables X₁, …, Xₙ with suitably bounded variances,

\[ \mathsf{P}\left\{ \left| \frac{X_{1} + \cdots + X_{n}}{n} - \frac{\mathsf{E}X_{1} + \cdots + \mathsf{E}X_{n}}{n} \right| > \epsilon \right\} \rightarrow 0 \]

as n → ∞ for any ε > 0. His method combined a rigorous formulation of the properties of mathematical expectation with the Chebyshev inequality, which yields the estimate n⁻²ε⁻² Σ D X_k for this probability.<sup>[12](https://encyclopediaofmath.org/wiki/Law_of_large_numbers)</sup>

In the modern textbook form, if X₁, …, Xₙ are independent with common mean μ and common variance σ², the sample mean has variance σ²/n, so Chebyshev's inequality gives

\[ \mathsf{P}\bigl(|\bar{X}_{n} - \mu| > t\bigr) \leq \frac{\sigma^{2}}{n\,t^{2}} \rightarrow 0, \]

which proves that the sample average converges in probability to μ.<sup>[4](https://faculty.washington.edu/yenchic/20A_stat512/Lec3_Expectation.html)</sup><sup> • </sup><sup>[13](https://www.cs.cornell.edu/courses/cs2800/2017sp/lectures/lec17-chebychev.html)</sup> More generally, the inequality shows that any sequence of random variables with a common mean and vanishing variance converges in probability to that mean.<sup>[4](https://faculty.washington.edu/yenchic/20A_stat512/Lec3_Expectation.html)</sup> Later proofs of the law of large numbers, including Markov's extension to variables without variance and Khinchin's 1929 theorem, are developments of Chebyshev's method.<sup>[12](https://encyclopediaofmath.org/wiki/Law_of_large_numbers)</sup>

## Sharpness and related inequalities

**The bound is exact.** For every σ² > 0 and every k ≥ σ there is a distribution with variance σ² for which P(|X − μ| ≥ k) equals σ²/k² exactly: a symmetric three-point distribution placing probability σ²/(2k²) at each of μ − k and μ + k and the remaining mass at μ. Consequently no inequality of the form P(|X − μ| ≥ k) ≤ c·σ²/k² with c < 1 can hold for all distributions; the constant 1 is optimal.<sup>[5](https://physics2u.com/maths/vault/chebyshev-inequality.html)</sup> The extremal distribution is not accidental: a linear program with three equality constraints has an optimal solution with at most three nonzero variables, which forces the three-point shape.<sup>[14](https://www.matpic.com/essays/the-bound-is-the-answer-to-a-search/)</sup>

**One-sided version.** Cantelli's inequality bounds a single tail: for a standardized variable with mean 0 and variance 1, P(Z ≥ v) ≤ 1/(1 + v²), and the bound is sharp.<sup>[15](https://link.springer.com/article/10.1007/s11749-022-00844-9)</sup> The naive one-sided use of Chebyshev gives σ²/k², and since σ²/(σ² + k²) < σ²/k² for every k > 0, Cantelli is strictly better, and unlike σ²/k² it is a genuine probability bound below 1 for all k > 0.<sup>[5](https://physics2u.com/maths/vault/chebyshev-inequality.html)</sup>

**Unimodal version.** If the distribution is known to be unimodal, the Vysochanskij–Petunin inequality improves the bound to P(|X − μ| ≥ tσ) ≤ 4/(9t²) for t > √(8/3).<sup>[5](https://physics2u.com/maths/vault/chebyshev-inequality.html)</sup>

**Position among concentration inequalities.** Chebyshev belongs to a family of concentration inequalities that bound the deviation of a random variable from its mean. It needs the mean and variance but neither independence nor bounded variables, and its tail decays as O(1/t²). Markov's inequality needs only the mean and decays as O(1/t); Chernoff and Hoeffding bounds require independence and bounded variables but decay exponentially as O(e^(−t²)), and Bernstein's bound is tighter still when the variance is small. Chebyshev is the tool of choice when variables may be dependent and only variance information is available.<sup>[6](https://courses.cs.washington.edu/courses/cse493s/25au/concentration-inequalities.pdf)</sup> For example, for the number of heads in n unbiased coin flips, Chebyshev gives P(X ≥ 3n/4) ≤ 4/n, improving the Markov bound of 2/3 by using the variance.<sup>[8](https://cs.uwaterloo.ca/~r5olivei/courses/2025-spring-cs466/lecture-notes/lecture3/)</sup>

## Relation to the empirical rule

The empirical rule (68–95–99.7) states precise coverage figures but holds only for normal distributions. Chebyshev's theorem makes a weaker but universally applicable statement: for any distribution with finite mean and standard deviation, at least 1 − 1/k² of observations fall within k standard deviations of the mean. At k = 2 this gives at least 75% against the normal distribution's approximately 95%, and at k = 3 at least 88.9% against approximately 99.7%. Chebyshev gives a floor; the true percentage can be much higher.<sup>[16](https://statohub.com/empirical-rule/)</sup>

## Applications in measurement

In metrology, the inequality supplies distribution-free confidence statements. The ISO Guide to the Expression of Uncertainty in [Measurement](https://www.edgepedia.org/measurement) and its NIST adaptation report a measurement as y ± k·u_c(y), where u_c(y) is the combined standard uncertainty and k a coverage factor. When nothing can be assumed about the distribution of the measurand except its mean and standard deviation, the Chebyshev inequality applies, and the coverage probability of the interval [y ± k·u_c(y)] is at least 75% for k = 2 and at least 89% for k = 3, whatever the distribution. If the distribution can be assumed symmetric and unimodal, the Gauss inequality raises these to at least 89% and 95% respectively.<sup>[17](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication500-244.pdf)</sup> The price of this generality is width: Chebyshev confidence intervals are conservative because they must accommodate any possible probability density. At 95% confidence the required Chebyshev factor is 4.472 against the normal distribution's 1.960, a 2.28-fold widening, and at 99% the factors are 10.000 against 2.575.<sup>[18](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir4720.pdf)</sup>

## References

1. [Chebyshev inequality in probability theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Chebyshev_inequality_in_probability_theory)
2. [Concentration Inequalities, G. W. Thomas, Stanford University notes](https://ai.stanford.edu/~gwthomas/notes/concentration.html)
3. [A Distribution-Independent Bound on the Level of Confidence in the Result of a Measurement, W. T. Estler, J. Res. NIST 102, 587 (1997)](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=820841)
4. [UW STAT 512, Lecture 3: Expectation and basic asymptotic theories](https://faculty.washington.edu/yenchic/20A_stat512/Lec3_Expectation.html)
5. [Chebyshev's inequality, maths2u](https://physics2u.com/maths/vault/chebyshev-inequality.html)
6. [Concentration Inequalities: Quick Reference, UW CSE 493S](https://courses.cs.washington.edu/courses/cse493s/25au/concentration-inequalities.pdf)
7. [Variance and spread, Libre University](https://en.libre.university/lesson/probability/variance-and-spread)
8. [Concentration Inequalities, R. Oliveira, University of Waterloo lecture notes](https://cs.uwaterloo.ca/~r5olivei/courses/2025-spring-cs466/lecture-notes/lecture3/)
9. [Sharp Inequalities of Bienaymé–Chebyshev and Gauß Type for Possibly Asymmetric Intervals around the Mean, arXiv:2208.08813](https://arxiv.org/abs/2208.08813)
10. [Bienaymé, Irenée-Jules, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Bienaym%C3%A9%2C_Iren%C3%A9e-Jules)
11. [A Tricentenary history of the Law of Large Numbers, E. Seneta, arXiv:1309.6488](https://arxiv.org/abs/1309.6488)
12. [Law of large numbers, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Law_of_large_numbers)
13. [Chebychev's inequality and weak law of large numbers, Cornell CS 2800, Spring 2017](https://www.cs.cornell.edu/courses/cs2800/2017sp/lectures/lec17-chebychev.html)
14. [The bound is the answer to a search, Matpic](https://www.matpic.com/essays/the-bound-is-the-answer-to-a-search/)
15. [link.springer.com](https://link.springer.com/article/10.1007/s11749-022-00844-9)
16. [Empirical Rule in Statistics: 68-95-99.7, Statohub](https://statohub.com/empirical-rule/)
17. [An interpretation of the Guide to the Expression of Uncertainty in Measurement, NIST Special Publication 500-244](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication500-244.pdf)
18. [Confidence intervals for discrete approximations to ill-posed problems, NISTIR 4720](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir4720.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Expectation, moments, and inequalities › Markov- and Chebyshev-type inequalities*

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