Chebyshev's Inequality states that for a random variable with finite variance, the probability that it deviates from its mean by more than k standard deviations is at most one divided by k squared. Named for Pafnuty Chebyshev, it applies to any distribution with finite variance regardless of shape, and it underlies the standard proof of the weak law of large numbers.
Facts
StatementChebyshev's inequality states that for a random variable with finite nonzero variance, the probability the variable deviates from its mean by more than k standard deviations is at most 1 divided by k squared, for any positive real number k. 1 Connections
Sources
1. Chebyshev's Inequality (Wikipedia)
Wikimedia Foundationlead section, first paragraph
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.
History section
The theorem was first proved by Bienaymé in 1853 and more generally proved by Chebyshev in 1867.
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