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A central composite design augments a factorial portion with axial points $\pm \alpha e_j$ and replicated centre points. It supports a full quadratic response model: intercept, linear terms, cross-products and squared coordinates. Centre-point replication estimates pure error. Choosing the axial distance appropriately gives a rotatable design.

For the initial normal linear model, the design matrix has rows $(1,x_1,x_2,x_3)$. The eight factorial corners have balanced signs and orthogonal coordinate columns; the six centre points contribute only to the intercept. ConsequentlyUsing the least-squares normal equations gives the least-squares estimatorIt is a linear transformation of independent normal responses, so its sampling distribution isThe four estimated coefficients are independent, since they are jointly normal with diagonal covariance matrix.

Adding the axial settings produces a central composite design: eight factorial corners, six axial points and six replicated centre points. The axial distance $1.682$ is the rounded value of $8^{1/4}=1.6817928\dots$. With that exact distance it is a rotatable design for a full quadratic model. Indeed, sign symmetry annihilates odd moments, all coordinates have the same second moment, and the fourth moments satisfyRotational symmetry requires the first to be three times the second, giving ${\alpha }^4=8$. Using the printed rounded distance makes rotatability approximate, rather than mathematically exact.

The full quadratic regression mean has ten coefficients:The expanded design matrix has rank ten. To verify this rather than just count parameters, suppose such a quadratic vanishes at every setting. Its centre value sets its constant term to zero; subtracting the values at opposite axial points sets every linear coefficient to zero, and adding them then sets every squared-coordinate coefficient to zero. On the factorial corners, orthogonality of the three pair-product columns forces their coefficients to zero as well. Thus all ten columns are independent, leaving $20-10=10$ residual statistical degrees of freedom.

There are $15$ distinct settings: eight corners, six axial points and the centre. The six centre replicates give the only pure error, with $6-1=5$ degrees and sum of squares $859.33$. The saturated setting-mean model has $15$ parameters, so the lack of fit beyond the quadratic model has $15-10=5$ degrees. For any setting $r$ with $n_r$ replicates, the identitysplits the residual sum of squares into pure error and lack of fit; the cross term vanishes because the deviations from the setting mean sum to zero. Summing givesUnder a correct quadratic mean with independent errors of constant normal variance, the pure-error and lack-of-fit projections are orthogonal, hence their sums of squares are independent with respective distributions ${\sigma }^2{\chi }_5^2$ and ${\sigma }^2{\chi }_5^2$. The Lack-of-fit F-test therefore usesThis is below the given $10\%$ and $5\%$ critical values $3.45$ and $5.05$. There is no significant lack of fit at either level. That is failure to reject this quadratic mean, not proof that it is exact or that a maximum has already been located. The pure-error denominator also assumes the error variance at the centre represents the common variance throughout the experiment.