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Taken from "An Introduction to Quantum Physics" by Stefanas Trachanas (Question 2.4).

I need to prove that for an arbitrary statistical distribution:

$$ \Delta A ≤ \frac{|a_{max}-a_{min}|}{2} $$

Where $a_{min}$ & $a_{max}$ are the minimum & maximum possible values of the distribution,respectively.

I have proved this for 2 special cases: The case in which the mean is at the centre of the range, & the case with only two possible outcomes. How do I prove the general case?

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Prove that the mean square deviation from any value is minimal if that value is the mean. Thus, the standard deviation is at most the root mean square deviation from the centre of the range. The distance from the centre of the range is bounded by the right-hand side of your inequality.

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