When can standard deviation be equal to range

Question

I was learning basic statistics and I read a proof that the standard deviation is always less than or equal to the range. I don't understand how range can ever be equal to the standard deviation. I could only think of one case: when all data points have the same value. Is there any other case possible in which the equality holds?

Answer 1

No, there is not.

The proof is a little boring and technical, if you care. Let $r$ denote the range, and assume $r>0$ (to rule out the case where all data points have the same value). Let $a$ denote the smallest value that can occur. Then $a+r$ must be the largest value that can occur (since the range is $r$), so all values are in the interval $[a,a+r]$. Let $\mu$ denote the mean. We must have $a \le \mu \le a+r$. It follows there must exist some value $x_0$ such that $|x_0-\mu| < r$ and that $x_0$ can occur in this distribution (in particular, if $a<\mu$, you can take $x_0=a+r$; if $\mu<a+r$, you can take $x_0=a$; since $r>0$, at least one of these two cases must hold). Moreover, for all other values of $x$, we have $|x-\mu| \le r$ (since $x \in [a,a+r]$ and $\mu \in [a,a+r]$). Therefore,

$$\mathbb{E}[|x-\mu|^2] < r,$$

since it is an average of numbers, one of which is $<r$ and all of the others of which are $\le r$. This expectation is exactly the S.D., so it proves that the S.D. is $<r$. That completes the proof that, apart from the one case you identified, the S.D. is always strictly less than the range.