Chebyshev's inequality

Bound on probability of a random variable being far from its mean

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 k σ {\displaystyle k\sigma } is at most 1 / k 2 {\displaystyle 1/k^{2}} , where k {\displaystyle k} is any positive constant and σ {\displaystyle \sigma } 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.

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