TITLE: Intuition behind Variance formula
QUESTION [11 upvotes]: Variance is given as:
$\operatorname{Var}(X) = \mathbb{E}[(X-\mathbb{E}(X))^2]$.
Is there an intuition behind this and can you find this formula starting from the second generating moment ?

REPLY [7 votes]: We want to measure the expected “deviation” of a random variable from it's expected value.  What comes first in mind is the expected difference  between the r.v.  and it's expected value, that is $E\bigl(X-E(X)\bigr)$.  But this number is always zero. --
Next, as drhab pointed out, is the average distance, i.e., the absolute value of the difference, between the r.v. and it's expected value, that is $E\bigl(|X-E(X)|\bigr)$. Now that appears to be a “natural” measure of the deviation, but there is a big drawback: it's not everywhere differentiable, so we can't apply Analysis to it.
Now we're stuck; what to do know?  Well, let's follow an idea of a great mathematician, in our case Gauss.  He introduced in his early years the expected value of the squared difference between the r.v. and it's expected value, that is $E\bigl((X-E(X))^2\bigr)$, aka variance of $X$.  Its drawback: it isn't intuitive. 
At the other hand, as  Stefanos wrote: “The moment of the most use is when k=2.”  That's right, but why is it the moment of most use?  The main reason for this is doubtlessly Chebyshev's inequality: http://en.wikipedia.org/wiki/Chebyshev%27s_inequality
And this inequality is very intuitive.