Why do RMS and Standard Deviations sum squares? I've seen some references to this - or a few examples of it, but for some reason it's not clicking. Can anyone give me a good layman's explanation of why calculations such as RMS values and Standard Deviations insist upon squaring all the values (before summing them) - which then means you have to take the square root of the sums to get a value of practical usefulness.
I wish I could articulate the questions a bit better - I know I've just essentially stated the definition of what an RMS value is - but why is this "better" then averaging absolute values? (Same goes for Standard Deviation).
This type of calculation also appears in the Pythagorean theorem - so I am sure the answer is somehow related to that. Bonus points if you can tie in your explanation to that, too ;-)
 A: If you tried to calculate the average distance data is away from the population mean $\mu$ you will find that the positive differences cancel with the negative ones.
To be specific $\sum (x_i-\mu)=0$ 
This shouldn't be that surprising because that's why one reason why we liked the mean in the first place.  The squaring first essentially turns all the negative distances from the mean into positives so the sum is no longer zero.  
You can avoid the problem using the modulus function and the result is called a mean deviation. I believe that squared values are just easier to work with than modulus.
This is a simplistic explanation and I'm sure there are many more sophisticated.
Here is a link to a discussion of measures of spread
A: 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.
