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I would like to know how we can determine the maximum value of standard deviation, $\sigma$.

Given a range for a population or sample, $[a, b]$ where $b$ is the maximum value and $a$ is the minimum value, I suppose there is also a range of $\sigma$ which I denote as $[0, m]$, i.e. $0 \leq \sigma \leq S$. I think the minimum $\sigma$ has to be 0 because if all samples are the same value then there is no deviation from the mean. But what about the maximum $\sigma$ from the mean? Essentially I want to know how to calculate the lower and upper bounds for $\sigma$, given the range of the population or sample.

It would be great if I could have some reference to a paper or book to validate the answers I get.

Hamman Samuel
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    see [here](http://stats.stackexchange.com/questions/45588/variance-of-a-bounded-random-variable#comment88534_45588) and [here](http://math.stackexchange.com/questions/83046/maximum-of-the-variance-function-for-given-set-of-bounded-numbers) – user603 Jun 22 '13 at 13:49
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    @user603 Thanks! If I understand the two links correctly, then the maximum of the standard deviation, S = (b-a)/2? – Hamman Samuel Jun 22 '13 at 14:00
  • yes. . . . . . . – user603 Jun 22 '13 at 14:16
  • As far as a reference goes, it's mentioned in section 4 of [this paper](http://www.stat.tugraz.at/AJS/ausg093/093Al-Saleh.pdf) – Glen_b Jun 22 '13 at 14:30

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