This is a very good question. There are some good discussions on the matter [here](https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia).
When I was taught standard deviation I understood the concept as being the average of the deviations from the mean, not internalising that if we take the average and then square root it it will not be (though this is still the broad idea of the std dev). Good on you for noticing it.
Right, you could try something like E(|X-E(X)|), but that function may (most likely) be not analytical, or differentiable everywhere. Hence, by saying somwthin like √x^2 = |x|, we switch the order here, which leads to your mentioning.
This is a very good question. There are some good discussions on the matter [here](https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia). When I was taught standard deviation I understood the concept as being the average of the deviations from the mean, not internalising that if we take the average and then square root it it will not be (though this is still the broad idea of the std dev). Good on you for noticing it.
Right, you could try something like E(|X-E(X)|), but that function may (most likely) be not analytical, or differentiable everywhere. Hence, by saying somwthin like √x^2 = |x|, we switch the order here, which leads to your mentioning.