Imagine I have a distribution like the following

![enter image description here][1]

*File [SkewedDistribution.png](https://en.wikipedia.org/wiki/File:SkewedDistribution.png) of Wikimedia Commons by [User:Audriusa](https://commons.wikimedia.org/wiki/User:Audriusa) licensed under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/deed.en)*

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

**My attempt:** I cannot use the [skewness](https://en.wikipedia.org/wiki/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?

  [1]: https://i.sstatic.net/ZP936.png