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I get that the variance of a random variable X is the expected value of the squared deviation from its mean, and the standard deviation is just the square root of that. Can you interpret the standard deviation as the absolute average deviation from the mean? Is the variance defined as such because if we defined the standard deviation as the expected value of X - E(X), we would get 0?

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The average deviation from the mean is $0$, that is $E[X-\mu]=0$, and so, taken literally, the absolute average deviation is also $0$. Changing your question slightly to

Can you interpret the standard deviation as the average absolute deviation from the mean?

No, $E[|X-\mu|]$ is not the standard deviation, that is, $$\sigma = \sqrt{E[(X-\mu)^2]} \neq E[|X-\mu|].$$

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The standard deviation is at least as large as the mean absolute deviation: The function $\varphi(z) = z^2$ is strictly convex over non-negative argument values; from Jensen's inequality it follows that:

$$\mathbb{V}(X)=\mathbb{E}\Big((X-\mu)^2\Big) = \mathbb{E}\Big(\varphi(|X-\mu|)\Big) \geqslant \varphi\Big(\mathbb{E}(|X-\mu|)\Big) = \mathbb{E}(|X-\mu|)^2,$$

with the inequality being strict unless $\mathbb{V}(|X-\mu|)=0$. It follows that $\mathbb{S}(X) \geqslant \mathbb{V}(|X-\mu|)$, with the inequality being strict unless $\mathbb{V}(|X-\mu|)=0$. So as you can see, unless it is equal to zero (i.e., for a random variable with a point mass distribution), the standard deviation is strictly greater than the mean absolute deviation.

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    $\begingroup$ Repost of an earlier comment with some deletions of incorrect assertions. I don't think this answer is correct. For $\sigma>0$, consider a discrete random variable $X$ that takes on values $\mu\pm\sigma$ with equal probability $\frac 12$. Then, $\mathbb V(X)=\mathbb S(X)=\sigma>0$. $\endgroup$ – Dilip Sarwate Jul 2 '18 at 14:30
  • $\begingroup$ -1 because of what @DilipSarwate wrote. Please ping me when/if you edit and I will happily remove the downvote. $\endgroup$ – amoeba Jul 4 '18 at 12:59
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    $\begingroup$ @amoeba: Condition for strict inequality now corrected. $\endgroup$ – Ben Jul 4 '18 at 13:49
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    $\begingroup$ I still don't understand how the bold sentence ("Unless the standard deviation is zero, it is strictly greater than mean absolute deviation") is true, given Dilip's example. In his example standard deviation is not zero. $\endgroup$ – amoeba Jul 4 '18 at 13:58
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    $\begingroup$ Changed -1 to +1 :) $\endgroup$ – amoeba Jul 5 '18 at 5:14

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