Different positive/negative error values

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If you have some distribution, the standard deviation is defined as symmetric about the mean; what measure do people use for different positive and negative error values?
 
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What do you mean? Why can't you use standard deviation?
 
What is it, exactly, that you are trying to measure?
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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