|mean - median| ≤ stddev

Dependencies:

1. Expected value of a random variable
2. Variance of a random variable
3. X ≤ Y ⟹ E(X) ≤ E(Y)
4. Linearity of expectation
5. Minimizer of f(z) = E(|X-z|) is median

$\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Var}{\operatorname{Var}}$ Let $X$ be a real-valued random variable. Let $m$ be a median of $X$. Then $(\E(X) - m)^2 \le \Var(X)$.

Proof

\begin{align} & -|X-m| \le X-m \le |X-m| \\ &\implies |\E(X-m)| \le \E(|X-m|) \\ &\implies \E(X-m)^2 \le \E(|X-m|)^2 \end{align}

Let $\mu = \E(X)$. Let $Y = |X-\mu|$. \[ \Var(Y) = \E(Y^2) - \E(Y)^2 = E((Y - \E(Y))^2) \ge 0. \] Hence, $\E(Y^2) \ge \E(Y)^2$, i.e., $\E((X-\mu)^2) \ge \E(|X-\mu|)^2$.

\begin{align} (\E(X)-m)^2 &= \E(X-m)^2 \tag{linearity of expectation} \\ &\le \E(|X-m|)^2 \\ &\le \E(|X-\mu|)^2 \tag{$m$ minimizes $f(z) = \E(|X-z|)$} \\ &\le \E((X-\mu)^2) = \Var(X) \end{align}

Dependency for: None

Info:

• Depth: 9
• Number of transitive dependencies: 25

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. /measure-theory/linearity-of-lebesgue-integral
5. /measure-theory/lebesgue-integral
6. σ-algebra
7. Generated σ-algebra
8. Borel algebra
9. Measurable function
10. Generators of the real Borel algebra (incomplete)
11. Measure
12. σ-algebra is closed under countable intersections
13. Group
14. Ring
15. Field
16. Vector Space
17. Probability
18. Random variable
19. Median of a random variable
20. Random variables: multiple medians
21. Expected value of a random variable
22. X ≤ Y ⟹ E(X) ≤ E(Y)
23. Linearity of expectation
24. Minimizer of f(z) = E(|X-z|) is median
25. Variance of a random variable