ninja1024

the Jarque–Bera test is a simple stats test for normality based on the first 4 moments, along the lines of Bchad's recommendation but you'll need to compute the JB test statistic and compare it against a table rather than eye-balling skewness and kurtosis. There is plenty of info about it on the web. There are also other more involved statistical tests which can also be implemented easily just with a spreadhseet, just google "tests for normality"

the reason why you want to use log-return is because it is additive, i.e. the log-return over some time interval is equal to the sum of the log-returns over a partition of that time interval. under certain set of assumptions, the sum of a large number of i.i.d. random variables will be approximately gaussian (CLT). so the assumption that the log-return is normal is grounded in theory, it is not just an empirical exercise about finding the best distribution that fits your data

brainsX

How many data points do you have? Just use a formula for the curve that has one less variable. It almost always fits well.

tango_gs

It's ln changes, not ln levels (in case you didn't get that).
It took a long time for me to understand why to use one or the other. In my opinion, there are two reasons beyond what Mobius says.
First, if you use natural logs and then transfer it to arithmetic, then the arithmetic will never fall below 0 (this is important for indices that never will fall below 0). Alternately, it may also be important that as the index goes to 0, then the volatility might fall.
Second, when you're doing like a mean-variance optimization, then you need to do it on arithmetic returns. If the frequency of your data matches your time horizon, this isn't an issue. However, if you have a longer time horizon, then you need to project out data and its tricky to project out arithmetic returns. B/c of what Mobius says, you can just add up the log returns and convert to arithmetic (exp(X)-1 does the trick). You can't use the log returns in the optimization b/c log returns don't add up the way arithmetic returns do.