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This is a fundamental question, though I didn't really get the meaning/idea what does it mean by aromaticity or degree of aromaticity. Does it mean the ability of a molecule to be aromatic?

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    $\begingroup$ Welcome to Chemistry.SE! Could you provide a little more context? I suspect that this term refers to the amount of aromatic resonance stability compared to benzene. For example, furan (despite meeting the Huckel requirements) is less aromatic, evidenced by its ability to act as a diene in a Diels-Alder reaction $\endgroup$ – Ben Norris Jan 19 '14 at 11:51
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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)
J. Am. Chem. Soc., 124, 13495 (2002)
Chem. Rev. 101, 1301 (2001)
Tetrahedron 57, 3689 (2001)

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

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I'm currently studying this and...

Degree of aromaticity can be assessed by many methods, both thermodynamic and spectroscopic but one of the favoured these days is actually computational. One of the characteristics of an aromatic system is that it gives rise to ring currents which then cause significant deshielding of the aromatic protons. This can be exploited by a technique called “Nuclear Independent Chemical Shift calculations”, or NICS. In this computational method, the shielding parameter (the CS bit) for an arbitrary dummy nucleus (the NI bit) in the centre of the ring is calculated. If the shielding parameter is negative, then the ring is aromatic, if the shielding parameter is positive, it is anti-aromatic and if zero, it’s just not a delocalised system.

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