We report on the measurement and statistical study of thousands of current-voltage characteristics and transition voltage spectra (TVS) of single-molecule junctions with different contact geometries that are rapidly acquired using a new break junction method at room temperature. This capability allows one to obtain current-voltage, conductance voltage, and transition voltage histograms, thus adding a new dimension to the previous conductance histogram analysis at a fixed low-bias voltage for single molecules. This method confirms the low-bias conductance values of alkanedithiols and biphenyldithiol reported in literature. However, at high biases the current shows large nonlinearity and asymmetry, and TVS allows for the determination of a critically important parameter, the tunneling barrier height or energy level alignment between the molecule and the electrodes of single-molecule junctions. The energy level alignment is found to depend on the molecule and also on the contact geometry, revealing the role of contact geometry in both the contact resistance and energy level alignment of a molecular junction. Detailed statistical analysis further reveals that, despite the dependence of the energy level alignment on contact geometry, the variation in single-molecule conductance is primarily due to contact resistance rather than variations in the energy level alignment.