Chi square test on non-normal distributions I'm a beginner in statistics and would like to understand one of the requirements for the chi-square test to work. It is that the random variable being measured has to follow a normal distribution.
My question then is, what other models should I use if the random variables I'm measuring do not follow a normal distribution? This is similar to how the Student-t test should be used instead of the Standard Normal for a small sample, is there an equivalent for chi-square test for a small sample?
 A: Normality is a requirement for the chi square test that a variance equals a specified value but there are many tests that are called chi-square because their asymptotic null distribution is chi-square such as the chi-square test for independence in contingency tables and the chi square goodness of fit test.  Neither of these tests require normality. This agrees with Peter Ellis' comment. 
Regarding your question when specific parametric assumptions are not made (normality being just one such assumption) there are nonparametric procedures (rank tests, permutation tests and the bootstrap) that can be applied sith more generality.  In regression, robust regression is an alternative to ordinary least squares.
A: I learned the chi squared distribution as a special case of a gamma density function.  What I have read today online on wikipedia and in texts sometimes says "If Z1, ..., Zk are independent, standard normal random variables, then the sum of their squares is distributed according to the chi-squared distribution with k degrees of freedom."  and then in other instances drops the "normal random variables" part- 
"By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k." Perhaps this is an error in wikipedia.
But the normal assumption is really the part I am interested in today.  It seems to me that there are NO "requirements for the chi-square test to 'work'" except maybe that the set of random variables not be empty, and be real numbers.  
By test I take the asker to mean the squaring of the sum of the squares of the random variables, and then checking that against what would be expected from a standard normal with the given mean and std dev.   That is to say the expected versus the observed outcome.  Here is a list of so called "chi-squared tests": http://en.wikipedia.org/wiki/Chi-squared_test 
The outcomes of a M = Z^2 where Z is a standard normal random variable are different than if M = G^2 where G was a random variable from a gamma distribution. 
An example I can think of is in application when there is a small sample size- I suppose this can be defined as less than the amount that a sample size calculator would yield- Here you don't know if your normal assumption is valid, because let's say you have no prior data, and the sample size being small means no central limit theorem application, but not all is lost because a chi-squared test can be done to measure the validity of Gaussian distribution functions being used such as normal probability distribution function, normal cumulative distribution function and their inverses etc.  
SO as far as I can tell one of the most useful uses of the chi-squared test in beginning and intermediate practice of statistics is to test the normal assumption on small sample sizes.
But to get to the part of the question that asks about other data types.  I think it is good to learn all the different distributions other than normal, and t distribution.  There are discrete probability distribution functions- Bernoulli, binomial etc. and there are continuous probability distribution functions- exponential, beta, Poisson, Pareto etc.  
Then learn about what a gamma distribution is- how it is an all encompassing distribution function.  From there simply looking at the data, graphing the data and measuring observed versus expected in some way i.e. "goodness of fit" etc can help determine what kind of distribution shape your data has.  What's great about a gamma or even exponential distribution is that you can make any shape you see.
Due to the central limit theorem and the other versions of that that there are- people often just assume a normal distribution.  And this is fine over many many tests and or much prior data.
This has been edited from the original comment.
