How to measure the difference of a distribution being normally distributed

Question

Imagine I have a distribution like the following

File SkewedDistribution.png of Wikimedia Commons by User:Audriusa licensed under CC-BY-SA 3.0

Now I want to measure, how this distribution differs from being normally distributed. What can I do?

My attempt: I cannot use the skewness because this only measures the symmetry of my distribution. Another way would be to calculate $\inf_{N\in\mathcal N} d(X,N)$ whereby $\mathcal N$ is the set of all normal distributions (i.e. the set of all distributions with density function $\frac{1}{\sigma\sqrt{2\pi} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$) and $d$ is a metric for distributions. Is this a good choice for measuring the difference of being normally distributed?

Answer 1

Start with a Normal probability plot https://en.wikipedia.org/wiki/Normal_probability_plot . The tails are where non-Normaility can be most extreme. If the probability calculations you are going to make on the distribution are roughly in the "middle" of the distribution, as opposed to the tails, you can get away with a lot. But if you're trying to do risk analysis, say, and need to understand the probability (risk) of an extreme outcome, the tails of the distribution (and one of those tails in particular) is all that matters, and that is where non-Normality (effect on probability calculation results) is usually by far the most extreme. Aside from being perhaps non-symmetric, most real-world distributions have much fatter tails than a Normal, no matter what variance or standard deviation is used, and therefore use of a Normal distribution often vastly understates the true probability of tail events (more than 2 or 3 standard deviations from the mean). Of course, dependence vs. independence across random variables or events can also be a huge driving factor in what the actual probability of an extreme event is.

So, along the same lines as whuber's comment, you have to base your assessment of closeness to Normality on why you care how close the distribution is to being Normally distributed and what you're going to do with the distribution.