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For a two-factor factorial design with coded corners $(\pm 1,\pm 1)$, observe each of the four corners once and add $n_0\ge 2$ independent center runs at $(0,0)$. Assume independent Gaussian errors with common variance ${\sigma }^2>0$. Write ${\overline{Y}}_F$ for the corner average, ${\overline{Y}}_0$ for the center average, and $S_0^2$ for the unbiased center-run sample variance. Then the center-point curvature contrast isFor the full quadratic regression mean ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_{12}{x}_1{x}_2+{\beta }_{11}{x}_1^2+{\beta }_{22}{x}_2^2$, the contrast has expected value ${\beta }_{11}+{\beta }_{22}$. The corner average cancels the linear and cross-product terms, while both squared coordinates equal one at every corner and zero at the center.

Under the null ${\beta }_{11}+{\beta }_{22}=0$,The independent normal contrast and center residuals justify this exact F-distribution; a first-order mean model alone would not imply it without the error assumptions. The center observations supply pure error, while the extra corner-to-center comparison detects one aggregate curvature direction.^[ref]

Zero contrast does not exclude curvature. The mean $x_1^2-x_2^2$ has a saddle shape but gives the same mean at all four corners and the center. A central composite design adds axial runs that can distinguish the separate squared terms.