Chebyshev's inequality

In probability theory, Markov’s inequality tells us that for a non-negative random variable $X$:

$$P(X\ge a)\le \frac{E(X)}{a}$$

i.e., we can construct a strict upper bound with only the expected value. This is usually a poor upper bound, so it’s not very helpful in practice.

Chebyshev’s inequality is a better upper bound, with no requirement of the signedness of the random variable. It states:

$$P(|Y-m|\ge b)\le \frac{\text{Var}(Y)}{b^2}$$

The way the inequality is formatted means we have to rewrite any probabilities (like $P(X\ge 99)$) into a Chebyshev form.