What does it mean by degree of aromaticity of a molecule?
Aromaticity might be viewed as resonance stabilzation energy, NMR shift, or altered reactivity.  Here is a way of measuring its change among related systems,

Chem. Commun.  48, 8144 (2012) 
J. Am. Chem. Soc. 131, 189 (2009) 
J. Am. Chem. Soc. 127, 16251 (2005) 
J. Org. Chem. 67, 8798 (2002) 
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http://nptel.ac.in/courses/IIT-MADRAS/Engineering_Chemistry_III/pdf/1_Aromaticity.pdf 
A more practical approach derives from MO theory and Hund's rule of maximum multiplicity.

http://www.youtube.com/watch?v=U6FX5RfLjd0 
http://www.youtube.com/watch?v=H0so_NJ2aVw 
Frost circle