math-mode
I'm trying to get a sum to display "properly" within a fraction. My input is:
\begin{equation}
E = 1 - \dfrac{\sum_{i=1}^{n}(O_{i}-P_{i})^{2}}{\sum_{i=1}^{n}(O_{i}-\bar{O})^{2}}
\end{equation}
And the output looks like this: 
I want the indices to appear above and below the sum as opposed to the side. How can I go about doing this?
Would something as simple as a \scalebox do?
\documentclass{article}
\usepackage{fixltx2e,graphicx,mathpazo}
\begin{document}
\( \sqrt{\scalebox{0.75}[1.0]{\( - \)}\frac{1}{9}} \)
% cf.
\( \sqrt{-\frac{1}{9}} \)
\end{document}
Here, I am using the optional argument to \scalebox to set the vertical scaling to 1, so that only the horizontal size changes.
I made your code compile and also replaced the arrays by matrices where I thought this would be appropriate. Notice also that there are environments like align that allow you to write aligned multiline equations. So here is some modification of your code that can be compiled and has some minor corrections.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
Let
\[
X =\begin{pmatrix}
1 & X_1\\
\dots & \dots\\
1 & X_n
\end{pmatrix}
\begin{pmatrix}
10 & 0\\
0 & 5
\end{pmatrix}
\]
Then,
\[
X^T =
\begin{pmatrix}
1 & \dots & 1\\
X_1 & \dots & X_N
\end{pmatrix}
\]
and
\begin{align*}
X \times X^T &=
\begin{pmatrix}
1 & X1\\
\dots & \dots\\
1 & X_n
\end{pmatrix}\,
\begin{pmatrix}
1 & \dots & 1\\
X_1 & \dots & X_N
\end{pmatrix} \\
&=
\begin{pmatrix}
n & \sum_{i}^n X_i\\
\sum_{i}^n X_i & \sum_{i}^n X_i^2
\end{pmatrix}
\end{align*}
as well as
\begin{align*}
(X^TX)^{-1}&=
\frac{1}{n\sum_{i=1}^n X_i^2 - \left(\sum_{i=1}^n X_i\right)^2}
\begin{pmatrix}
\sum_{i=1}^n X_i^2 & -\sum_{i=1}^n Xi\\
- \sum _{i=1}^n X_i & n
\end{pmatrix} \\
&=\frac{1}{n \sum _{i=1}^n X_i^2 - ( \sum X_i)^2}
\left[n \sum_{i=1}^n X_i^2 - \left( \sum _{i=1} ^n X_i\right)^2\right]
\\
&=1
\end{align*}
\end{document}
It is not hard to see, though, that the first equation does not make too much sense in view of the equations below it. On the other hand, I did not just dare to remove the right-most matrix, but I am sure you will know what to do with it.
Best Answer
Use
\limitsjust before_and^in connection with\sumcommand, the same holds also for an integral\int\limits^{b}_{a}for example.