What is standard variation and why is it the square root of the variance?

Question

anonymous · Answer

the standard deviation is a special type of mean,
it measures  the dispersion of items 
ie. how far numbers are moving away from the mean
$$x_{i}-mean$$ 

but since some numbers can deviate negetively the result has to be squared.
the average of the squared result is the varience
$$\sum(x-mean)^2/n$$
the sd is the square root of the varience because those number were initially unsquared 
we squared them to avoid adding negetives

anonymous · Answer

What would this be in terms of bell curves? Surely you would have to factor in the probability at x along with how far it deviates from x, some thing like $$P(x) (x-m)^2$$

anonymous · Answer

Also, I take it that the SD is the square root of the variance rather than the variance the square of the SD (ie- the Standard deviation is an absolute value also)?