For the second order estimating function, ŷ(x), and third order true function η(x) in two variables, secondary criteria are used to select a design from the class of average minimum bias designs. It is shown that designs can be constructed which not only allow estimation of a cubic model with nearly minimum variance averaged over a square but also provide for efficient lack of fit tests. Average qiradratic variance is also evaluated. Specific 12 point and 16 point designs are recommended.