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Little Bluestem
Yes, the SD can be greater than the mean, even when there is a cut-off at zero. The catch is that this cannot be true of a normal (Gaussian) distribution, as I have argued for some time. In a normal distribution, the mean, median and mode are essentially the same, and the only thing you can know about it beyond mean and SD is the number of samples. It contains absolutely no other information.
If this is not true, the obvious inference is that the distribution is not normal, and
common parametric statistics do not apply.
I can explain this to some people with mathematical training in terms of limitations on the application of the
Central Limit Theorem, but I suspect a group which already has 5 PhDs is incapable of learning, even without the presence of an MD.
All the proofs of which I am aware depend on linear combinations of many distributions, which has a hidden assumption that distributions will combine additively. If distributions combine multiplicatively, as in reliability calculations where a single failure can destroy an entire system, the resulting distribution will not be normal.
Lévy distributions are one class of example, and these fail to have a well-defined SD. The SD you compute will be an artifact of the bounds you set and the number of samples. This is not a good prerequisite for ANOVA, unless you are determined to commit a deception, safe in the knowledge that few will go to the trouble to check your hidden assumptions.