Conceptual questions dealing with chi-square distributions? In my textbook they have this inequality:
$$ \chi_{1-\frac{\alpha}{2}}^2 < \frac{(n-1)s^2}{\sigma^2} < \chi_{\frac{\alpha}{2}}^2$$
which later becomes this statement:
$$\frac{(n-1)s^2}{\chi_{\frac{\alpha}{2}}^2 } < \sigma^2 < \frac{ (n-1)s^2}{ \chi_{1-\frac{\alpha}{2}}^2}$$
Now I know the whole idea is to find the confidence interval for $\sigma^2$ the variance, but I was wondering if the distribution for the variance is normal. I also don't understand why the chi square is squared. 
When I look at the picture in the book that shows a right skewed graph with the chi squares labeled (i.e. $\chi_{0.95}^2 = 4.575$ and $\chi_{0.05}^2 = 19.675$), I get the impression that I'm looking at something similar to $z$ scores. What are these chi squares? Do they represent the number of standard deviations away from the mean?
 A: 
I was wondering if the distribution for the variance is normal.

No, as the above calculation demonstrates, sample variances from independent and identically distributed normal data have a scaled chi-square distribution ... which is to say, a gamma distribution.
However, as the degrees of freedom in the $\chi^2$ grow very large, that starts to look more and more normal.

I also don't understand why the chi square is squared.

If you're asking "Why is the chi-square written as the square of the symbol $\chi$ rather than some symbol without a square?" that's basically an old convention that dates back to Karl Pearson, and derives from a notation he used when working with particular quadratic forms in the multivariate normal. 
Just think of $\chi^2$ as a single symbol rather than the square of something (it is the square of something, but that something is not especially interesting when dealing with this sort of problem).

When I look at the picture in the book that shows a right skewed graph with the chi squares labeled (i.e. $χ^2_{0.95}=4.575$ and $χ^2_{0.05}=19.675$), I get the impression that I'm looking at something similar to z scores. What are these chi squares?

Those values are the places on a particular chi-square density (one with 11 df) that have 95% and 5% of the probability (area) to their right, respectively. They are akin to quantiles of the normal distribution ($Z_{\alpha}$ values), yes, but for a $\chi^2_{11}$ distribution instead of a standard normal.

Do they represent the number of standard deviations away from the mean?

No
