Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

In 2 recent Statistics Lessons of the Day, I

Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar.

$P[|X - \mu| \geq k \sigma] \leq 1 \div k^2$

Markov’s inequality roughly says that a random variable $X$ is most frequently observed near its expected value, $\mu$.  Remarkably, it quantifies just how often $X$ is far away from $\mu$.  Chebyshev’s inequality goes one step further and quantifies that distance between $X$ and $\mu$ in terms of the number of standard deviations away from $\mu$.  It roughly says that the probability of $X$ being $k$ standard deviations away from $\mu$ is at most $k^{-2}$.  Notice that this upper bound decreases as $k$ increases – confirming our intuition that it is highly improbable for $X$ to be far away from $\mu$.

As with Markov’s inequality, Chebyshev’s inequality applies to any random variable $X$, as long as $E(X)$ and $V(X)$ are finite.  (Markov’s inequality requires only $E(X)$ to be finite.)  This is quite a marvelous result!