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Given the range $R$ ($= max - min$) of $n$ data points ($x_1, x_2, ..., x_n$), what is the range of these points' standard deviation $\sigma$? It's easy to see that the min value of $\sigma$ can be $0$ (when all data points are of the same value), but what is the max value of $\sigma$, and why? My intuition is when the $n$ points are separated equally (e.g., $\frac{0}{n-1}R, \frac{1}{n-1}R, \frac{2}{n-1}R, ..., \frac{n-1}{n-1}R$), $\sigma$ may be the largest (which is $\sqrt{\frac{n+1}{12(n-1)}}R$ if I did my math right). But I am not sure. Anyone can prove this, or disprove this and show and prove the true maximum value of $\sigma$?

Besides (more generally), this is the question for the 1-dimensional data points. I am also curious about the result on higher dimensional data points. It is very welcome if you can suggest some ideas/proofs.

Daniel
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  • https://stats.stackexchange.com/questions/45588/variance-of-a-bounded-random-variable/93493#93493 fully answers your question: just take the square root of the variance. It's unclear what you mean by "higher dimensional data points:" what would be the analogs of $R$ and $\sigma$? – whuber Jul 07 '21 at 16:07

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Taking $n/2$ points to be the minimum and $n/2$ to be the maximum gives a standard deviation of $$\sqrt{R^2\frac{1}{2}\left(1-\frac{1}{2}\right)}=\frac{R}{2}$$ (for even $n$)

Thomas Lumley
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  • For all even $n$, your result is $\ge$ my result. Two more questions: (1) whether is this the maximum value for even $n$? (2) what if $n$ is odd? – Daniel Jul 07 '21 at 07:47
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    Yes, it's the maximum value for even $n$. For odd $n$ I think the maximum is with nearly half the points at one end and nearly half at the other end, but it's a bit more annoying to prove – Thomas Lumley Jul 07 '21 at 07:49
  • For even $n$, can you prove this is the maximum value? – Daniel Jul 07 '21 at 09:01
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    The duplicate proves it for even $n.$ For odd $n,$ the theory of Lagrange multipliers shows all values must be at the endpoints of the interval, leaving just $(n+1)/2$ cases to examine (which is straightforward algebra). – whuber Jul 07 '21 at 16:14