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Chebyshev's inequality

Chebyshev's inequality (or Tchebysheff's inequality) is a result in statistics that gives a lower bound for the probability that a value of a random variable with finite variance lies within a certain distance from the variable's mean; equivalently, the theorem provides an upper bound for the probability that values lie outside the same distance from the mean. The theorem applies even to non "bell-shaped" distributions and puts bounds on how much of the data is or is not "in the middle".

Theorem. Let X be a random variable with mean μ and finite variance σ2. Now, for any real number k > 0,

Only the cases k > 1 provide useful information.

For illustration, assume Wikipedia articles are on average 1000 characters long with a standard deviation of 200 characters. From Chebyshev's inequality we can then deduce that at least 75% of Wikipedia articles have a length between 600 and 1400 characters (k = 2).

Another consequence of the theorem is that for any distribution with mean μ and finite standard deviation σ, at least half of the values lie in the interval (μ-√2 σ, μ+√2 σ).

The bounds provided by Chebyshev's inequality cannot, in general, be improved upon; it is possible to construct a random variable where the Chebyshev bounds are exactly equal to the true probabilities. Typically, however, the theorem will provide rather loose bounds.

The theorem can be useful despite these loose bounds because it applies to a wide variety of variables, including those that are nothing close to normally distributed, and because the bounds are easy to calculate.

Chebyshev's inequality is used for proving the weak law of large numbers.

The theorem is named in honor of Pafnuty Chebyshev.

A one-tailed variant with k > 0, is

See also: Markov's inequality.