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 😉
Best Answer
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