Summation arithmetic clarification Suppose I have an expression as follows:
$= \frac{(\sum_{i = 1}^N (x_i - \bar{x}))^2}{(\sum_{i = 1}^N (x_i - \bar{x})^2)^2} $
$= \frac{1}{\sum_{i = 1}^N (x_i - \bar{x})^2}$
My question is, how did $\frac{(\sum_{i = 1}^N (x_i - \bar{x}))^2}{(\sum_{i = 1}^N (x_i - \bar{x})^2)^2} $ simplify to $\frac{1}{\sum_{i = 1}^N (x_i - \bar{x})^2}$? I know that $\sum a_i^2 \neq (\sum a_i)^2$, but the expression simplified $\frac{(\sum a_i)^2}{(\sum a_i^2)^2}$ to $\frac{1}{\sum a_i^2}$? Am I missing something here?
 A: Without knowing what $\bar x$ indicates, it is not easy to provide a precise answer. In general, we could note that the initial equation of the OP 
$$ \frac{(\sum_{i = 1}^N (x_i - \bar{x} ))^2}{(\sum_{i = 1}^N (x_i - \bar{x})                   ^2)^2} = \frac{1}{\sum_{i = 1}^N (x_i - \bar{x})^2}   $$
can be written as 
$$ (\sum_{i = 1}^N (    x_i - \bar{x}    ))^2 = \frac{(\sum_{i = 1}^N (  x_i - \bar{x}     )^2)^2}{\sum_{i = 1}^N (x_i - \bar{x})^2}   $$
and then 
$$ (\sum_{i = 1}^N (   x_1 - \bar{x}    ))^2 = \sum_{i = 1}^N (    x_i - \bar{x}  )^2 $$
The last equation is generally not true. In particular, the equation is valid in the limit case that the summations contain only one element (as in this case both sides become $(x_1 - \bar{x})^2   \ $) but is no longer valid if the summations contain $\geq 2$ elements. For example, if $N=2$, the equation becomes
$$( x_1 - \bar{x}  + x_2  - \bar{x}  ) ^2=  (x_1 - \bar{x} )^2 + (x_2  - \bar{x} )^2$$
which leads to
$$( x_1 - \bar{x}) ( x_2  - \bar{x}) =0$$
So, for N=2, the initial equation of the OP is valid only if  $x_1 = \bar{x}$ and/or $x_2 = \bar{x} \ $ (note that these conditions lead to the case where the summations have only one element). 
For higher values of $N $, the condition that must be satisfied to make the initial equation valid has the form
$$ \sum_{i = 1}^{N-1}  \sum_{j = {i+1}}^N  (x_i - \bar{x}) (x_j - \bar{x}  )=0$$
Again, however, the initial equation is generally not true. 
