How to calculate the charge distribution at the ortho/para or meta sites of a substituted benzene ring
Summary: The method used does not handle π-conjugation

The partial charges in Avogadro (and many other software programs) are assigned by default by the Gasteiger-Marsili scheme†

Importantly, while the Gastieger charges are fairly good for atomic partial charges of organic-ish molecules, they're unlikely to reproduce effects in conjugated systems very well.


  A method is presented for the rapid calculation of atomic charges in σ-bonded and nonconjugated π-systems. Atoms are characterized by their orbital electronegativities. In the calculation only the connectivities of the atoms are considered. Thus only the topology of a molecule is of importance.


My suggestion is to use some type of quantum chemical calculation, e.g.:


NWChem
OpenMOPAC
ORCA
(etc.)


Even semiempirical quantum chemical methods (e.g. PM7) will allow you to assign charges based on the quantum electrostatic potential around the molecule. They'll be much more accurate at the trends you want.

† "Iterative Partial Equalization of Orbital Electronegativity - A Rapid Access to Atomic Charges," Tetrahedron, Vol. 36, pp. 3219-3228, 1980.