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For a real random variable $X$ with finite variance ${\sigma }^2$, expected value $\mu$, and a threshold $a>0$, Chebyshev's inequality givesOne can also cap this upper bound at one, since a probability cannot exceed one. If $\sigma >0$, taking $a=k\sigma$ gives $\mathbb{P}(|X-\mu |\ge k\sigma )\le 1/k^2$. For example, at least $3/4$ of the probability lies strictly within two standard deviations of the mean. This statement needs only the finite variance assumption. If the variance is zero, $X=\mu$ almost surely and every positive deviation threshold has zero tail probability.

Consider a random variable uniformly distributed on $\{-2,-1,0,1,2\}$. Its expected value is zero and its variance is $(4+1+0+1+4)/5=2$. At $a=2$ the actual tail probability is $2/5$, while Chebyshev's inequality gives $2/2^2=1/2$. At $a=3$ the actual tail probability is zero, while the bound is $2/9$. A valid bound need not describe the exact shape or bounded support of a particular probability distribution.

For the standard normal distribution, the actual tail probabilities beyond two and three standard deviations are approximately $0.0455003$ and $0.0026998$, whereas the general bounds are $1/4$ and $1/9$. More knowledge of the probability distribution can therefore improve the estimate. The comparison does not diminish the guarantee: the three-point example in equality in Chebyshev's inequality attains it exactly.

The proof is a one-line use of Markov's inequality on the nonnegative random variable $Y=(X-\mu {)}^2$. The event $|X-\mu |\ge a$ equals $Y\ge a^2$, and $\mathbb{E}Y={\sigma }^2$. This explains why the denominator is squared and why a finite variance supplies the necessary information. ^[ref].

One important use is control of a sample mean. For independent and identically distributed random variables with finite variance ${\sigma }^2$, the variance of a sample mean is ${\sigma }^2/n$. HenceFor every fixed tolerance, the bound tends to zero as the sample size grows, proving the finite-variance case of the weak law of large numbers. For ${\sigma }^2=1$ and $\epsilon =0.1$, $n=2000$ makes the bound $0.05$. This is a sufficient bound under the stated assumptions, rather than an optimal required sample size.

An early historical source is Pafnuty Chebyshev's 1867 paper “Des valeurs moyennes”, published in French translation by N. de Khanikof. Its opening theorem bounds deviations of sums using their expected values and second moments, linking averages with control of exceptional outcomes. ^[ref].

Pafnuty Chebyshev's 1867 paper “Des valeurs moyennes” develops bounds for deviations of sums in terms of their expected values and second moments. This connects his historical work with the modern Chebyshev's inequality and its use in the weak law of large numbers. ^[ref].