In several introductory statistics books, for a list of data $X = \{x_1,\ldots,x_n\}$, I have frequently seen the following rule-of-thumb:
$$ s \approx \frac{\text{range}(X)}{4} = \frac{\max(X)-\min(X)}{4} $$
I have thought about this a little and can offer my heuristics, but I am curious if this (obviously very crass) rule-of-thumb can be put on slightly more stable mathematical footing.
Suppose the data are normally distributed. Then approximately $95\%$ of the data should be within two standard deviations of the mean, in which case $4s \approx \text{range}(X)$ is plausible.
For a continuous uniform distribution on $(a,b)$, the exact value is $s = (b-a)/\sqrt{12}$, compared to the approximation $(b-a)/4$. Since $1/\sqrt{12}\approx 0.2887$, the relative error is about $15.47\%$; not great, not terrible. This value is in the same ballpark for the discrete normal distribution.
Looking at another distribution, the Poisson distribution with mean $\mu>0$ and standard deviation $\sqrt{\mu}$, this amounts to summing over values $k$ with $|\mu-k|\le 2 \sqrt{\mu}$; this becomes weird because of rounding since $k$ can only take integer values, but a plot produced about $0.92$ as a minimum value for this sum.
The book did use this rule-of-thumb as a way to approximate the range of a population given a sample (of which the range and standard deviation can be calculated), but I think the reason for the heuristic in the first place is to provide an alternative to students who do not have the tech or computational skills to find the exact value of $s$ in the first place. Any further justification of this heuristic would be appreciated.