Its interesting that even in this thread the 2 answers differ by an order of magnitude lol
In reality, it depends on your actual distribution, but the OP from this thread here is unreasonably conservative for something described as a "rule of thumb". Almost always, if you have at least 10 of every category, you can already discover every interesting thing that a rule of thumb will allow. And you probably could go with less. But if you want precision, you can't get it with rules of thumb.
The exponential bit applies to the probability densities as a function of the bounds themselves, i.e. how likely you are to fall x units away from the mean typically decreases exponentially with (some polynomial in) x.
Of course, this is all assuming a whole bunch of standard conditions on the data you're looking at (independence, identically distributed, bounded variance, etc.) and may not hold if these are violated.