Chebyshev's inequality
In probability theory,
Chebyshev's inequality guarantees that in any probability distribution, "nearly all" values are close to the mean — the precise statement being that no more than 1/
k2 of the distribution's values can be more than
k standard deviations away from 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 completely arbitrary distributions, for example it can be used to prove the weak law of large numbers. In practical usage, in contrast to the empirical rule, which applies to normal distributions, under Chebyshev's inequality 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 the Markov's inequality, especially in the context of analysis.