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I am learning about standard deviation. In the book I am reading it defines standard deviation as:

$$\sigma^2 = E[(x - E[x])^2]$$

so far no problem. In then states "simple mathematical manipulation leads to the following equation:"

$$\sigma^2 = E[x^2] - (E[x])^2$$

Where I understand the second term is the (squared) expected value of the experiment. My problem is that (as simple as the author says the manipulations are), I do not know how to go from the first equation to the second. What would be the steps that allow him to write the second equation starting from the first? What's the proof for this?

Your help is greatly appreciated, as always. Thank you.

Marc Ourens
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  • Just expand the inner bracket and use linearity of expectations. Also remember $E(X)$ is a number, not a random variable. – Macavity Sep 01 '13 at 20:12

1 Answers1

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Write $\mu = E[x]$. The expected value is additive, $E[f+g] = E[f] + E[g]$ and the expected value of a constant is that constant, $E[c] = c$, so you compute

$$\begin{align} E[(x-\mu)^2] &= E[x^2 - 2\mu x + \mu^2]\\ &= E[x^2] - 2\mu E[x] + E[\mu^2]\\ &= E[x^2] - 2\mu^2 + \mu^2\\ &= E[x^2] - \mu^2. \end{align}$$

Daniel Fischer
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  • Great fantastic. Thank you so much. I started writing this derivation down on paper, and just stopped in the middle ;-( Thank you. – Marc Ourens Sep 01 '13 at 20:14
  • For future reference there is a good video on Khan Academy explaining this as well: http://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/statistics--alternate-variance-formulas – Marc Ourens Sep 09 '13 at 08:33