Data from the UN and Germany shows that 80%+ of refugees have a university degree or secondary education. When war strikes, those with more resources are the ones able to escape and pay their way through to the EU.
Of course this doesn't mean integration is any easier, but the reasons are not that simple.
[1] - https://www.reuters.com/article/us-europe-migrants-germany-s...
http://www.e-ir.info/2016/03/07/refugees-economic-burden-or-...
Also I'm not sure if crime stats are right. There have been a lot of cases of covering up crimes by immigrants/refugees due to social pressure and allegations of racism. See the Rotheram incident, which set of a storm of reports all over Europe, but was covered up for years due to social pressure (56 mass rapists in one town, 53 of whom were immigrant Muslims): https://en.wikipedia.org/wiki/Rotherham_child_sexual_exploit...
Another example from Germany, 1200 assaults in one night in one town: https://en.wikipedia.org/wiki/New_Year%27s_Eve_sexual_assaul...
[1] - http://www.telegraph.co.uk/news/2017/04/25/migrant-crime-ger...
[2] - https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2903116