How about this

\documentclass{book}
\usepackage{amsmath}
\begin{document}
\begin{equation}
G_i(d) = \frac{\sum_j w_{ij} (d) x_j - W_i\bar{x}(i)}
{s(i)\{[(n - 1) S_{1i} - W_i^2]/{{(n - 2)}\}^{1/2}}},\qquad j\neq i.
\end{equation}
\end{document}
Following the comment, you could also try
\begin{equation}
G_i(d) = \frac{\displaystyle\sum_j w_{ij} (d) x_j - W_i\bar{x}(i)}
{s(i)\{[(n - 1) S_{1i} - W_i^2]/{{(n - 2)}\}^\frac{1}{2}}},\qquad j\neq i.
\end{equation}
which gives

The error I got with your code is
! Extra }, or forgotten $.
l.8 {s(i)\{[(n - 1) S_{1i} - W_i^2]}}
/{{(n - 2)}^{1/2}}, j\neq i.
?
which is one of TeX's more helpful errors. I find that in these situations it is helpful to copy the offending line, comment out the original, and then strip the copy down to a bare minimum. You can then build it back up slowly to try and avoid the errors.
One particularly useful piece of advice is always open and close grouping parenthesis/brackets/braces at the same time.