Q: Is data distributed as a Gaussian? I know there are methods to check whatever the observed data DOES NOT follow a Normal distribution. I'm thinking about chi-square test, Anderson test etc.
Is there any function/test/metric which says how Gaussian-like is the data distribution?
Thanks

A: Departures from the assumption of normality are not easy to quantify since they are not uniquely defined. For instance, the data may present departures from normality in terms of asymmetry, heavier tails, lighter tails, multimodality, et cetera.
The use of hypothesis tests does not provide evidence in favour of a hypothesis, for philosophical reasons.
There are some visual tools that you can use to see how close to normality your sample is, such as QQ-plots (against normal quantiles).
If you have a clue about the type of departure your data present (you can take a look at their histogram en 1-D), then you can try to fit a model that contains the normal distribution as a particular case, and see whether this is close to the normal distribution or not.  For instance, if your data seem to have heavier tails than normal and asymmetry, you could use some sort of skew-t distribution to fit your data (or some other distribution that contain the normal distribution as a limit case or a particular case). Then, take a look at the estimators of the degrees of freedom and the skewness parameter to see whether they indicate departures from the assumption of normality (e.g. if the estimator of the skewness parameter is very close to zero (or the one that reduces the model to symmetry) and the estimator of the degrees of freedom is large, then the data should look close to normal). You can also fit a normal model and calculate some distance/divergence between the flexible fitted model and the normal fitted model (e.g. Kullback–Liebler divergence, ...) 

A: The Jarque-Bera test checks for kurtosis and skew that match the normal distribution. 
http://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test
