Two practical methods of improving the accuracy of gaussian statistical linearization are compared (Beaman 1981 a, Leithead 1990 a). Both result in an improved estimate for the covariance of the systems. Three examples are used to compare the improved estimates with the gaussian estimate and, where possible, the exact solution. The first example is a scalar odd non-linearity driven by zero mean white noise. The second is a second-order system with odd symmetry and thus zero steady-state mean. The final example is a first-order system which has a bias introduced to produce a non-zero output. It is shown that both methods are an improvement over the gaussian method, but the method of Leithead is computationally easier than that of Beaman. The methods are then applied to the design of a control system for a position servomechanism with a backlash.