I think I've figured out the answer:
A "mean deviation" (as described in the link Karl provided) would do an accurate job if your sample set very evenly or flatly distributed - as in a very "square" shaped dataset like (9,9,9,10,10,10,11,11,11) has an intuitive Mean Absolute Deviation ("MAD") of 2/3rds. It's easy to see that the mean of this dataset is 10, and the average deviation from this is 2/3rd - as a third of the values are the mean, 1/3rd is one below, and 1/3 is one above. So for this kind of "squarely" distributed dataset, the MAD is an intuitive representation of the "width" of the square, or "spread" of the sample values.
However, the concept of Standard Deviation is specifically built on the assumption that populations will follow a Bell-Shaped (Gaussian) curve, which itself is created with a function like like/related to an RMS value (even called a "Gaussian RMS width").
So for my purposes - in which my samples in no-way follow a "normal distribution" - simply stating what the "standard deviation" is fairly meaningless. (In my case, it's much larger than the mean itself!) Likewise, if I had a very square distribution, the "MAD" would be descriptive. If I did have a "normal distribution" the "SD" would be descriptive.
While I'm not a mathematician (or even close) - this somewhat explains my question about the relationship between SD and RMS.
I would assume the relation to the pythagorean theorem lines within the similarities to the shape of the bell-curve and that of a sine-wive, which is a derivation of how a circle is plotted, etc.