probability variance formula intuition Here is the formula for the variance : $\sigma^2=\dfrac{\sum(X-\mu)^2}{N}$.
My question is why do we SQUARE the difference between the mean and the variable, why don't we use absolute value ...The result will be different!
I saw online that it's to make the value positive, but like why don't we choose 4 or 6 as exponent? 
Hope someone can enlighten me! 
Thanks :)
Ayman

Here's a reason (not necessarily the reason). You can use either of the following measures of deviation from a given point $x$, i.e. how closely a set of values $x_1,...,x_n$ are grouped round a given point $x$ (smaller values being more closely):


*

*$\frac1n\sum_{i=1}^n(x_i-x)^2$ or

*$\frac1n\sum_{i=1}^n|x_i-x|$.


How would you use either of these measures to determine how closely the set of values are grouped together (i.e. if you don't yet have a specific value $x$ you're interested in, just the values $x_i$)? The natural thing to do is to say we'll work out how closely grouped around the best choice $x$ they are, i.e. the choice of $x$ which makes the deviation as small as possible. For option 1 (variance), the best value of $x$ is always the mean of the set of numbers (this is easy to check by differentiating wrt $x$). That means mean and variance naturally go together - if you wanted a measure for how close to the mean the values are, variance would be it.
But for option 2, the best value of $x$ to take is the median. So median and "average absolute deviation" go together like mean and variance. Option 2 isn't something you work with very often, simply because the median isn't used anywhere near as often as the mean. 
