# Main difference between standard deviation and average deviation?

I'm currently working on exam review for an upcoming statistics exam and I've managed to dig myself to far into the theoretical background of basic statistical principles. I'm currently looking at the definition of variance and standard deviation.

I understand that when given a sample, lets say $S=\{s_1, s_2, s_3, s_4, ...., s_n\}$ from an observation I can calculate the following $$d(S,\bar{S})=\sqrt{(s_1 - \bar{S})^2 + (s_2 - \bar{S})^2 +\ ...\ + (s_n - \bar{S})^2} = \sqrt{\sum_{i=1}^n(s_i - \bar{S})^2}$$

which is the Euclidean distance or as I interpret it, the combined deviation of all my observations.

My question is then, is the standard deviation the same as the average deviation? Also, could someone explain to me the reasoning for dividing by $n - 1$ instead of $n$ without involving the use of moments? And why do we divide inside the squareroot?