Most often tests for normality are not robust (a presence of outliers affect the results and data significantly deviate from normality). On the other hand, robust methods of normality testing  can deal with it. Another article, very worthy of reading for everyone.
- M. Stehlík, L. Střelec, and M. Thulin, "On robust testing for normality in chemometrics", Chemometrics and Intelligent Laboratory Systems, vol. 130, pp. 98-108, 2014. http://dx.doi.org/10.1016/j.chemolab.2013.10.010