Chebyshev's Inequality

Chebyshev’s Inequality is used to give the Probability of a Sample being beyond/within $k$ standard deviations of the mean.

$$\frac{E[X]=\mu \text{Var}(X)={\sigma }^{2}k>0}{P(|X-\mu |\ge k\sigma )\le \frac{1}{k^2}P(|X-\mu |<k\sigma )\ge 1-\frac{1}{k^2}}$$

or equivalently

$$\frac{E[X]=\mu \text{Var}(X)={\sigma }^{2}k>0}{\max (P(|X-\mu |\ge k\sigma ))=\frac{1}{k^2}\min (P(|X-\mu |<k\sigma ))=1-\frac{1}{k^2}}$$