Chebyshev's Inequality Theorem

This is nice b/c we can use this with any distribution shape, not just bell-shaped distributions.

The proportion (percent or fraction) of values from a dataset that will fall within k standard deviations of the mean will be at least $((1-\frac{1}{z^2})\cdot 100)$, where z is a real number that has an absolute value greater than 1 (z is not necessarily an integer).

For discrete random variables, this means that we can end up with a heuristic for if a number falls within a certain place in a probability distribution.

$\ge 0$ chance it falls within $\sigma$ of $\mu$. $\ge \frac{3}{4}$ it falls within $\pm 2\sigma$ $\ge \frac{8}{9}$ it falls within $\pm 3\sigma$

Unlike the Empirical Rule, it doesn’t only work on bell-shaped distributions.