This paper reports the average Nusselt number for steady, laminar natural convection between a vertical surface and otherwise quiescent pseudoplastic and dilatant fluids under a constant and uniform surface heat flux boundary condition. Models for the fluids' apparent viscosity were utilized that are valid in all five regions of the flow curve. The results are thus applicable for whatever shear rates may exist within the flow field and a dimensionless shear rate parameter was identified that quantifies the shear rate region where the given system is operating. The data indicate that the average Nusselt numbers approach the corresponding Newtonian values when the shear rates are predominantly in either the zero or the infinite shear rate Newtonian regions. However, power law values are approached only when both of the following two conditions are met: (1) the shear rates are principally in the power law region and (2) the fluid's limiting zero and infinite shear rate Newtonian viscosities differ sufficiently, by approximately 4 orders of magnitude or more. For all other cases, the average Nusselt number was found to reside between the Newtonian and the power law asymptotes. Results are provided in both graphical and tabular form over a broad range of system parameters.