Mixed-mean velocities were computed by numerical integration using a comprehensive correlating equation for the velocity distribution, and independently using a corresponding correlating equation for the local turbulent shear stress. As contrasted with earlier analytical integrations for grossly simplified velocity distributions, these calculations take into account the effects of the boundary layer near the wall and the wake in the central region. The resulting sets of numerical values for the mixed-mean velocity are correlated almost exactly in terms of equations with a theoretically derived structure. On the basis of the generally overlooked analogy of MacLeod, the same distributions for the velocity and the turbulent shear stress are used for both round tubes and parallel plates, but of course, the expressions for the mixed-mean velocity differ because of the different areas of integration. The analysis reveals inter alia that the hydraulic-diameter and laminar-equivalent-diameter concepts are fundamentally unsound. The correlating equations for the mixed-mean velocity are extended for the transition engendered by naturally rough surfaces using the speculation of Colebrook, and are reexpressed directly in terms of the friction factor and the Reynolds number for convenience.