Standard Deviation Formula Derivation

\[\begin{aligned} \sigma^2 &\overset{\text{def}}{=}\frac{\sum f\left(x-\overline{x}\right)^2}{\sum f}\\ &=\frac{1}{\sum f}\sum f\left(x-\frac{\sum fx}{\sum f}\right)^2\\ &=\frac{1}{\sum f}\sum f\left[x^2-2x\left(\frac{\sum fx}{\sum f}\right)+\left(\frac{\sum fx}{\sum f}\right)^2\right]\\ &=\frac{1}{\sum f}\left[\sum fx^2-\sum2fx\left(\frac{\sum fx}{\sum f}\right)+\sum f\left(\frac{\sum fx}{\sum f}\right)^2\right]\\ &=\frac{\sum fx^2}{\sum f}-2\left(\frac{\sum fx}{\sum f}\right)\left(\frac{\sum fx}{\sum f}\right)+\frac{\sum f}{\sum f}\left(\frac{\sum fx}{\sum f}\right)^2\\ &=\frac{\sum fx^2}{\sum f}-2\left(\frac{\sum fx}{\sum f}\right)^2+\left(\frac{\sum fx}{\sum f}\right)^2\\ \sigma^2 &=\frac{\sum fx^2}{\sum f}-\left(\frac{\sum fx}{\sum f}\right)^2\\ \sigma&=\boxed{\sqrt{\frac{\sum fx^2}{\sum f}-\left(\frac{\sum fx}{\sum f}\right)^2}} \end{aligned}\]

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$