In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of a random variable (with finite variance) from its mean. More specifically, the probability that a random variable deviates from its mean by more than is at most , where is any positive constant and is the standard deviation (the square root of the variance).
The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.88% within three standard deviations for a broad range of different probability distributions.
The term Chebyshev's inequality may also refer to Markov's inequality, especially in the context of analysis. They are closely related, and some authors refer to Markov's inequality as "Chebyshev's First Inequality," and the similar one referred to on this page as "Chebyshev's Second Inequality."
Chebyshev's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
History The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé. The theorem was first proved by Bienaymé in 1853 and more generally proved by Chebyshev in 1867. His student Andrey Markov provided another proof in his 1884 Ph.D. thesis.Markov A. (1884) On certain applications of algebraic continued fractions, Ph.D. thesis, St. Petersburg
Statement Chebyshev's inequality is usually stated for random variables, but can be generalized to a statement about measure spaces.
Probabilistic statement Let (integrable) be a random variable with finite non-zero variance (and thus finite expected value ).Feller, W., 1968. An introduction to probability theory and its applications, vol. 1. p227 (Wiley, New York). Then for any real number ,
Only the case is useful. When the right-hand side and the inequality is trivial as all probabilities are at most 1.
As an example, using shows that the probability values lie outside the interval does not exceed . Equivalently, it implies that the probability of values lying within the interval (i.e. its "coverage") is at least .
For the general case, for any ,
Because it can be applied to completely arbitrary distributions provided they have a known finite mean and variance, the inequality generally gives a poor bound compared to what might be deduced if more aspects are known about the distribution involved. Min. % within standarddeviations of meanMax. % beyond standarddeviations from mean10%100%50%50%1.555.56%44.44%275%25%287.5%12.5%388.8889%11.1111%493.75%6.25%596%4%697.2222%2.7778%797.9592%2.0408%898.4375%1.5625%998.7654%1.2346%1099%1%
Modification with strict inequality signs In case then for any positive real numbers and
respectively.
Measure-theoretic statement Let be a measure space, and let be an extended real-valued measurable function defined on . Then for any real number and ,
More generally, if is an extended real-valued measurable function, nonnegative and nondecreasing, with then:
This statement follows from the Markov inequality, , with and , since in this case . The previous statement then follows by defining as if and otherwise.
Example Suppose we randomly select a journal article from a source with an average of 1000 words per article, with a standard deviation of 200 words. We can then infer that the probability that it has between 600 and 1400 words (i.e. within standard deviations of the mean) must be at least 75%, because there is no more than chance to be outside that range, by Chebyshev's inequality. But if we additionally know that the distribution is normal, we can say there is a 75% chance the word count is between 770 and 1230 (which is an even tighter bound).
Sharpness of bounds As shown in the example above, the theorem typically provides rather loose bounds. However, these bounds cannot in general (remaining true for arbitrary distributions) be improved upon. The bounds are sharp for the following example: for any ,
For this distribution, the mean is and the variance is , so the standard deviation is and
Chebyshev's inequality is an equality for precisely those distributions which are affine transformations of this example.
Proof
Markov's inequality states that for any non-negative real-valued random variable and any positive number , we have . One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable with :
It can also be proved directly using conditional expectation:
Chebyshev's inequality then follows by dividing by . This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where is thrown away, and the lower bound of on the event can be quite poor.
Chebyshev's inequality can also be obtained directly from a simple comparison of areas, starting from the representation of an expected value as the difference of two improper Riemann integrals (at the drawing in the definition of expected value for arbitrary real-valued random variables).
Extensions Several extensions of Chebyshev's inequality have been developed.
Selberg's inequality Selberg derived a generalization to arbitrary intervals. Suppose is a random variable with mean and variance . Selberg's inequality states that if ,
When , this reduces to Chebyshev's inequality. These are known to be the best possible bounds.
Finite-dimensional vector
Chebyshev's inequality naturally extends to the multivariate setting, where one has random variables with mean and variance . Then the following inequality holds.
This is known as the Birnbaum–Raymond–Zuckerman inequality after the authors who proved it for two dimensions. This result can be rewritten in terms of vectors with mean , standard deviation , in the Euclidean norm .
One can also get a similar infinite-dimensional Chebyshev's inequality. A second related inequality has also been derived by Chen. Let be the dimension of the stochastic vector and let be the mean of . Let be the covariance matrix and . Then
where is the transpose of . The inequality can be written in terms of the Mahalanobis distance as
where the Mahalanobis distance based on is defined by
Navarro proved that these bounds are sharp, that is, they are the best possible bounds for that regions when we just know the mean and the covariance matrix of .
Stellato et al. showed that this multivariate version of the Chebyshev inequality can be easily derived analytically as a special case of Vandenberghe et al. where the bound is computed by solving a semidefinite program (SDP).
Known correlation
If the variables are independent this inequality can be sharpened.
Berge derived an inequality for two correlated variables . Let be the correlation coefficient between and and let be the variance of . Then
This result can be sharpened to having different bounds for the two random variablesLal D. N. (1955) A note on a form of Tchebycheff's inequality for two or more variables. Sankhya 15(3):317–320 and having asymmetric bounds, as in Selberg's inequality. Isii K. (1959) On a method for generalizations of Tchebycheff's inequality. Ann Inst Stat Math 10: 65–88
Olkin and Pratt derived an inequality for correlated variables.
where the sum is taken over the variables and
where is the correlation between and .
Olkin and Pratt's inequality was subsequently generalised by Godwin.Godwin H. J. (1964) Inequalities on distribution functions. New York, Hafner Pub. Co.
Higher moments
Mitzenmacher and Upfal note that by applying Markov's inequality to the nonnegative variable , one can get a family of tail bounds
For we obtain Chebyshev's inequality. For and assuming that the th moment exists, this bound is tighter than Chebyshev's inequality. This strategy, called the method of moments, is often used to prove tail bounds.
Exponential moment A related inequality sometimes known as the exponential Chebyshev's inequality Section 2.1 is the inequality
Let be the cumulant generating function,
Taking the Legendre–Fenchel transformation of and using the exponential Chebyshev's inequality we have
This inequality may be used to obtain exponential inequalities for unbounded variables. (the references for this article are corrected by )
Bounded variables If has finite support based on the interval , let , where is the absolute value of . If the mean of is zero then for all Dufour (2003) Properties of moments of random variables
The second of these inequalities with is the Chebyshev bound. The first provides a lower bound for the value of .
Finite samples
Univariate case Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known and may not exist, but the sample mean and sample standard deviation from samples are to be employed to bound the expected value of a new drawing from the same distribution. The following simpler version of this inequality is given by Kabán.
where is a random variable which we have sampled times, is the sample mean, is a constant and is the sample standard deviation.
This inequality holds even when the population moments do not exist, and when the sample is only weakly exchangeably distributed; this criterion is met for randomised sampling. A table of values for the Saw–Yang–Mo inequality for finite sample sizes () has been determined by Konijn. The table allows the calculation of various confidence intervals for the mean, based on multiples, C, of the standard error of the mean as calculated from the sample. For example, Konijn shows that for , the 95 percent confidence interval for the mean is , where (this is 2.28 times larger than the value found on the assumption of normality showing the loss on precision resulting from ignorance of the precise nature of the distribution).
An equivalent inequality can be derived in terms of the sample mean instead,
A table of values for the Saw–Yang–Mo inequality for finite sample sizes () has been determined by Konijn.
For fixed and large the Saw–Yang–Mo inequality is approximately
Beasley et al have suggested a modification of this inequality
In empirical testing this modification is conservative but appears to have low statistical power. Its theoretical basis currently remains unexplored.
Dependence on sample size The bounds these inequalities give on a finite sample are less tight than those the Chebyshev inequality gives for a distribution. To illustrate this let the sample size and let . Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. Kabán's version of the inequality for a finite sample states that at most approximately 12.05% of the sample lies outside these limits. The dependence of the confidence intervals on sample size is further illustrated below.
For , the 95% confidence interval is approximately ±13.5789 standard deviations.
For the 95% confidence interval is approximately ±4.9595 standard deviations; the 99% confidence interval is approximately ±14.00 standard deviations.
For the 95% confidence interval is approximately ±4.5574 standard deviations; the 99% confidence interval is approximately ±11.1620 standard deviations.
For the 95% and 99% confidence intervals are approximately ±4.5141 and approximately ±10.5330 standard deviations respectively.