Variance is Least Mean Square Deviation about Point

Theorem

Let $S$ be an object upon which:

the mean square deviation

the expectation

the variance

is defined.

For each point $x$ in $S$, let $\map M x$ denote the mean square deviation of $S$ about $x$.

Then the value of $x$ for which $\map M x$ is the minimum is the variance of $S$.

That is, the expectation $\bar x$ of $S$ is the value of $S$ for which the mean square deviation is smallest.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean squared error
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean squared error