A scanning tunnelling microscope has been used to determine the conductance of single molecular wires with the configuration X-bridge - X, X-bridge-Y and Y-bridge-Y (X = thiol terminus and Y = COOH). We find that for molecular wires with mixed functional groups (X - bridge - Y) the single molecule conductance decreases with respect to the comparable symmetric molecules. These differences are confirmed by theoretical computations based on a combination of density functional theory and the non-equilibrium Green's functions formalism. This study demonstrates that the apparent contact resistance, as well as being highly sensitive to the type of the anchoring group, is also strongly influenced by contact-asymmetry of the single molecular junction which in this case decreases the transmission. This highlights that contact-asymmetry is a significant factor to be considered when evaluating nanoelectrical junctions incorporating single molecules.