Proof: The third standardized moment equals the skewness
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The Book of Statistical Proofs ▷
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Third standardized moment is skewness
Metadata: ID: P528 | shortcut: momstand-3rd | author: JoramSoch | date: 2026-02-27, 11:52.
Theorem: The third standardized moment equals the skewness, i.e.
\[\label{eq:momstand-3rd} \mu_3^{*} = \mathrm{Skew}(X) \; .\]Proof: The third standardized moment of a random variable $X$ with expected value $\mu$ and standard deviation $\sigma$ is defined as
\[\label{eq:momstand-3rd-def} \mu_3^{*} = \frac{\mathrm{E}[(X-\mu)^3]}{\sigma^3}\]which is equivalent to the definition of the skewness:
\[\label{eq:momstand-3rd-qed} \mu_3^{*} = \mathrm{Skew}(X) \; .\]∎
Sources: Metadata: ID: P528 | shortcut: momstand-3rd | author: JoramSoch | date: 2026-02-27, 11:52.