In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean (or equivalently, at least 1 − 1/k2 of the distribution's values are within k standard deviations of the mean). 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.
In practical usage, in contrast to the 68–95–99.7 rule, which applies to normal distributions, Chebyshev's inequality is weaker, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 89% within three standard deviations.
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."
The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé.:98 The theorem was first stated without proof by Bienaymé in 1853 and later proved by Chebyshev in 1867. His student Andrey Markov provided another proof in his 1884 Ph.D. thesis.
Only the case is useful. When the right-hand side and the inequality is trivial as all probabilities are ≤ 1.
As an example, using shows that the probability that values lie outside the interval does not exceed .
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.
|k||Min. % within k standard
deviations of mean
|Max. % beyond k standard|
deviations from mean
More generally, if g is an extended real-valued measurable function, nonnegative and nondecreasing on the range of f, then
The previous statement then follows by defining as if and otherwise.
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 k = 2 standard deviations of the mean) must be at least 75%, because there is no more than 1⁄k2
= 1/ 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 k ≥ 1,
For this distribution, the mean μ = 0 and the standard deviation σ = 1/, so
Chebyshev's inequality is an equality for precisely those distributions that are a linear transformation of this example.
Proof (of the two-sided version)
Markov's inequality states that for any real-valued random variable Y and any positive number a, we have Pr(|Y| > a) ≤ E(|Y|)/a. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable Y = (X − μ)2 with a = (kσ)2.
It can also be proved directly. For any event A, let IA be the indicator random variable of A, i.e. IA equals 1 if A occurs and 0 otherwise. Then
The Inequality of direct proof shows why the bounds are quite loose in typical cases:
- If , instead of taking the indicating value 0 as given by the left side of the inequality, a positive value of is counted.
- If , instead of taking the indicating value 1 as given by the left side of the inequality, a value greater or equal to 1 is counted. In some cases it exceeds 1 by a very wide margin.
Fix and let be defined as , and let be the indicator function of the set . Then, it is easy to check that, for any ,
since g is nondecreasing on the range of f, and therefore,
The desired inequality follows from dividing the above inequality by g(t).
Proof assuming random variable X is continuous
Replacing kσ with ε, where k=ε/σ, we have another form of the Chebyshev's inequality:
or, the equivalent
where ε is defined the same way as k; any positive real number.
Several extensions of Chebyshev's inequality have been developed.
Asymmetric two-sided case
When the distribution is known to be symmetric for any
where σ2 is the variance.
Similarly when the distribution is asymmetric or is unknown and
where σ2 is the variance and μ is the mean.
Let X1, X2 be two random variables with means μ1, μ2 and finite variances σ1, σ2 respectively. Then
where for i = 1, 2,
Two correlated variables
There are now three cases.
- Case A: If and then
- Case B: If the conditions in case A are not met but k1k2 ≥ 1 and
- Case C: If none of the conditions in cases A or B are satisfied then there is no universal bound other than 1.
where Xi is the i-th random variable, μi is the i-th mean and σi2 is the i-th variance.
where the sum is taken over the n variables and
where ρij is the correlation between Xi and Xj.
Ferentinos has shown that for a vector X = (x1, x2, ...) with mean μ = (μ1, μ2, ...), standard deviation σ = (σ1, σ2, ...) and the Euclidean norm || ⋅ || that
where YT is the transpose of Y. A simple proof was obtained in Navarro as follows:
and is a symmetric invertible matrix such that: . Hence and where represents the identity matrix of dimension n. Then and
Finally, by applying Markov's inequality to Z we get
and so the desired inequality holds.
The inequality can be written in terms of the Mahalanobis distance as
where the Mahalanobis distance based on S is defined by
Navarroproved 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 X.
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).
There is a straightforward extension of the vector version of Chebyshev's inequality to infinite dimensional settings. Let X be a random variable which takes values in a Fréchet space (equipped with seminorms || ⋅ ||α). This includes most common settings of vector-valued random variables, e.g., when is a Banach space (equipped with a single norm), a Hilbert space, or the finite-dimensional setting as described above.
Suppose that X is of "strong order two", meaning that
for every seminorm || ⋅ ||α. This is a generalization of the requirement that X have finite variance, and is necessary for this strong form of Chebyshev's inequality in infinite dimensions. The terminology "strong order two" is due to Vakhania.
Let be the Pettis integral of X (i.e., the vector generalization of the mean), and let
be the standard deviation with respect to the seminorm || ⋅ ||α. In this setting we can state the following:
- General version of Chebyshev's inequality.
Proof. The proof is straightforward, and essentially the same as the finitary version. If σα = 0, then X is constant (and equal to μ) almost surely, so the inequality is trivial.
then ||X − μ||α > 0, so we may safely divide by ||X − μ||α. The crucial trick in Chebyshev's inequality is to recognize that .
The following calculations complete the proof:
An extension to higher moments is also possible:
Let K(t) be the cumulant generating function,
Taking the Legendre–Fenchel transformation of K(t) and using the exponential Chebyshev's inequality we have
Inequalities for bounded variables
If P(x) has finite support based on the interval [a, b], let M = max(|a|, |b|) where |x| is the absolute value of x. If the mean of P(x) is zero then for all k > 0
The second of these inequalities with r = 2 is the Chebyshev bound. The first provides a lower bound for the value of P(x).
Let 0 ≤ X ≤ M where M > 0. Then
- Case 1:
- Case 2:
- Case 3:
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 N samples are to be employed to bound the expected value of a new drawing from the same distribution.
where X is a random variable which we have sampled N times, m is the sample mean, k is a constant and s is the sample standard deviation. g(x) is defined as follows:
Let x ≥ 1, Q = N + 1, and R be the greatest integer less than Q/x. Let
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 (N < 100) 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 N = 59, the 95 percent confidence interval for the mean m is (m − Cs, m + Cs) where C = 4.447 × 1.006 = 4.47 (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).
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 N = 100 and let k = 3. 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 N = 10, the 95% confidence interval is approximately ±13.5789 standard deviations.
For N = 100 the 95% confidence interval is approximately ±4.9595 standard deviations; the 99% confidence interval is approximately ±140.0 standard deviations.
For N = 500 the 95% confidence interval is approximately ±4.5574 standard deviations; the 99% confidence interval is approximately ±11.1620 standard deviations.
For N = 1000 the 95% and 99% confidence intervals are approximately ±4.5141 and approximately ±10.5330 standard deviations respectively.
The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively.
Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. Samuelson's inequality states that all values of a sample will lie within √ standard deviations of the mean. Chebyshev's bound improves as the sample size increases.
When N = 10, Samuelson's inequality states that all members of the sample lie within 3 standard deviations of the mean: in contrast Chebyshev's states that 99.5% of the sample lies within 13.5789 standard deviations of the mean.
When N = 100, Samuelson's inequality states that all members of the sample lie within approximately 9.9499 standard deviations of the mean: Chebyshev's states that 99% of the sample lies within 10 standard deviations of the mean.
When N = 500, Samuelson's inequality states that all members of the sample lie within approximately 22.3383 standard deviations of the mean: Chebyshev's states that 99% of the sample lies within 10 standard deviations of the mean.
Stellato et al. simplified the notation and extended the empirical Chebyshev inequality from Saw et al. to the multivariate case. Let be a random variable and let . We draw iid samples of denoted as . Based on the first samples, we define the empirical mean as and the unbiased empirical covariance as . If is nonsingular, then for all then
As , the right-hand side tends to which corresponds to the multivariate Chebyshev inequality over ellipsoids shaped according to and centered in .
Chebyshev's inequality is important because of its applicability to any distribution. As a result of its generality it may not (and usually does not) provide as sharp a bound as alternative methods that can be used if the distribution of the random variable is known. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.
Let X be a random variable with finite variance Var(X). Let Z be the standardised form defined as
Cantelli's lemma is then
This inequality is sharp and is attained by k and −1/k with probability 1/(1 + k2) and k2/(1 + k2) respectively.
If k > 1 and the distribution of X is symmetric then we have
An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). The upper (σ+2) and lower (σ−2) semivariances are defined as
where m is the arithmetic mean of the sample and n is the number of elements in the sample.
The variance of the sample is the sum of the two semivariances:
Chebyshev's inequality can now be written
A similar result can also be derived for the upper semivariance.
If we put
Chebyshev's inequality can be written
Because σu2 ≤ σ2, use of the semivariance sharpens the original inequality.
If the distribution is known to be symmetric, then
This result agrees with that derived using standardised variables.
The result of this linear transformation is to make P(a ≤ X ≤ b) equal to P(|Y| ≤ k).
The mean (μX) and variance (σX) of X are related to the mean (μY) and variance (σY) of Y:
With this notation Selberg's inequality states that
where a ≥ 0.
The bound on the one tailed variant is known to be sharp. To see this consider the random variable X that takes the values
- with probability
- with probability
Then E(X) = 0 and E(X2) = σ2 and P(X < 1) = 1 / (1 + σ2).
- An application – distance between the mean and the median
The one-sided variant can be used to prove the proposition that for probability distributions having an expected value and a median, the mean and the median can never differ from each other by more than one standard deviation. To express this in symbols let μ, ν, and σ be respectively the mean, the median, and the standard deviation. Then
There is no need to assume that the variance is finite because this inequality is trivially true if the variance is infinite.
The proof is as follows. Setting k = 1 in the statement for the one-sided inequality gives:
Changing the sign of X and of μ, we get
As the median is by definition any real number m that satisfies the inequalities
this implies that the median lies within one standard deviation of the mean. A proof using Jensen's inequality also exists.
Let μ = 0 and σ2 be the variance. Let γ = E(X3)/σ3 and κ = E(X4)/σ4.
If k2 − kγ − 1 > 0 then
The necessity of k2 − kγ − 1 > 0 requires that k be reasonably large.
Mitzenmacher and Upfal's inequality
for any integer k > 0 and that
is the 2kth central moment. They then show that for t > 0
For k = 1 we obtain Chebyshev's inequality. For t ≥ 1, k > 2 and assuming that the kth moment exists, this bound is tighter than Chebyshev's inequality.
Several other related inequalities are also known.
where Mm is the m-th moment and σ is the standard deviation.
He, Zhang and Zhang's inequality
Let X be a random variable with a ≤ X ≤ b and E[X] = 0, then for any s > 0, we have
Van Zuijlen's bound
Let Xi be a set of independent Rademacher random variables: Pr(Xi = 1) = Pr(Xi = −1) = 0.5. Then
The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31).
If the mode is not zero and the mean (μ) and standard deviation (σ) are both finite, then denoting the median as ν and the root mean square deviation from the mode by ω, we have
Gauss' bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode due to the Vysochanskiï–Petunin inequality. The latter has been extended by Dharmadhikari and Joag-Dev
where s is a constant satisfying both s > r + 1 and s(s − r − 1) = rr and r > 0.
It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions.
Unimodal symmetrical distributions
The bounds on this inequality can also be sharpened if the distribution is both unimodal and symmetrical. An empirical distribution can be tested for symmetry with a number of tests including McWilliam's R*. It is known that the variance of a unimodal symmetrical distribution with finite support [a, b] is less than or equal to ( b − a )2 / 12.
Let the distribution be supported on the finite interval [ −N, N ] and the variance be finite. Let the mode of the distribution be zero and rescale the variance to 1. Let k > 0 and assume k < 2N/3. Then
If 2 / √ < k ≤ 2N / 3 the bounds are attained by the distribution
where βk = 4 / 3k2, δ0 is the Dirac delta function and where
The existence of these densities shows that the bounds are optimal. Since N is arbitrary these bounds apply to any value of N.
Effects of symmetry and unimodality
Symmetry of the distribution decreases the inequality's bounds by a factor of 2 while unimodality sharpens the bounds by a factor of 4/9.
Because the mean and the mode in a unimodal distribution differ by at most √ standard deviations at most 5% of a symmetrical unimodal distribution lies outside (2√ + 3√)/3 standard deviations of the mean (approximately 3.840 standard deviations). This is sharper than the bounds provided by the Chebyshev inequality (approximately 4.472 standard deviations).
These bounds on the mean are less sharp than those that can be derived from symmetry of the distribution alone which shows that at most 5% of the distribution lies outside approximately 3.162 standard deviations of the mean. The Vysochanskiï–Petunin inequality further sharpens this bound by showing that for such a distribution that at most 5% of the distribution lies outside 4√/3 (approximately 2.981) standard deviations of the mean.
Symmetrical unimodal distributions
For any symmetrical unimodal distribution
- at most approximately 5.784% of the distribution lies outside 1.96 standard deviations of the mode
- at most 5% of the distribution lies outside 2√/3 (approximately 2.11) standard deviations of the mode
DasGupta's inequality states that for a normal distribution at least 95% lies within approximately 2.582 standard deviations of the mean. This is less sharp than the true figure (approximately 1.96 standard deviations of the mean).
Bounds for specific distributions
- DasGupta has determined a set of best possible bounds for a normal distribution for this inequality.
- Steliga and Szynal have extended these bounds to the Pareto distribution.
- Grechuk et.al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. In particular, they derived Chebyshev inequality for distributions with log-concave densities.
When the mean (μ) is zero Chebyshev's inequality takes a simple form. Let σ2 be the variance. Then
With the same conditions Cantelli's inequality takes the form
The first inequality is sharp. This is known as the Paley–Zygmund inequality.
The value of C0 is optimal and the bounds are sharp if
then the sharp bound is
Integral Chebyshev inequality
If f and g are of opposite monotonicity, then the above inequality works in the reverse way.
One use of Chebyshev's inequality in applications is to create confidence intervals for variates with an unknown distribution. Haldane noted, using an equation derived by Kendall, that if a variate (x) has a zero mean, unit variance and both finite skewness (γ) and kurtosis (κ) then the variate can be converted to a normally distributed standard score (z):
This transformation may be useful as an alternative to Chebyshev's inequality or as an adjunct to it for deriving confidence intervals for variates with unknown distributions.
While this transformation may be useful for moderately skewed and/or kurtotic distributions, it performs poorly when the distribution is markedly skewed and/or kurtotic.
The Environmental Protection Agency has suggested best practices for the use of Chebyshev's inequality for estimating confidence intervals. This caution appears to be justified as its use in this context may be seriously misleading.
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