Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ Second standardized moment is one

Theorem: The second standardized moment is zero, i.e.

\[\label{eq:momstand-2nd} \mu_2^{*} = 1 \; .\]

Proof: The second standardized moment of a random variable $X$ with expected value $\mu$ and standard deviation $\sigma$ is defined as

\[\label{eq:momstand-2nd-def} \mu_2^{*} = \frac{\mathrm{E}[(X-\mu)^2]}{\sigma^2} \; .\]

By plugging in $\mu = \mathrm{E}(X)$ and with the definitions of variance and standard deviation, we have

\[\label{eq:momstand-2nd-qed} \begin{split} \mu_2^{*} &= \frac{\mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right]}{\sigma^2} \\ &= \frac{\mathrm{Var}(X)}{\left( \sqrt{\mathrm{Var}(X)} \right)^2} \\ &= \frac{\mathrm{Var}(X)}{\mathrm{Var}(X)} \\ &= 1 \; . \end{split}\]
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Metadata: ID: P527 | shortcut: momstand-2nd | author: JoramSoch | date: 2026-02-27, 11:46.