Why square the result of $x_1 - \bar{x}$ in the standard deviation? I don't understand the necessity of square the result of  $x_1 - \bar{x}$ in  $$\sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}}$$. In fact I don't understand even why is $N - 1$ on the denominator instead of just $N$. Someone could explain it or recommend a good text about it? All books about Errors Theory or even Statistics that I found are either too much abstract or too much simplist.
Thanks in advance.

Hint:  You can measure the spread by Minimum Absolute Deviation |(x-xbar)| or the squared differences.  Take an example of the sequence, -3,0,3.  Xbar for this is 0.  Assume that you don't measure the Absolute deviation, then the spread is 0 if you just took x-xbar.  The mean squared will avoid this situation and give you an objective measure of spread (to bring the unit of spread to the original measure, you take the square root of it.  As far as dividing by N-1, N-1 is the number of observations minus the the degrees of freedom(measure  of (number of estimators)).  Here you are calculating the x-bar which is an estimator and hence subtract one.
