1. Arithmetic mean of all the values of Modulus[X(mean) - x(i)]

2. Geometric mean of all the values of Modulus[X(mean) - x(i)]

These two methods also make sense. And, we don't have to do the square root in the end, because basically we're just finding the average of the differences. So, I'm talking about using Modulus instead of squaring to get rid of negative differences.

Actually, there's another one I could think of:

3.Why don't we just calculate all the values of [X(mean) - x(i)] raised to any even power n ( to get rid of negative differences), then find arithmetic mean of these values and then calculate the nth root of that arithmetic mean? Why do we just use n=2? Basically, the standard deviation formula would now look like:

(arithmetic mean ((X(mean) - x(i))^n))^(1/n) , where n is even. Why do we just use n=2?

So, what's wrong with these 3 formulas of standard deviation?