Independent slope and curvature estimates in a symmetric quadratic design ($\hat{\beta }=({\bar{Y}}_{+}-{\bar{Y}}_{-})/2,\hat{\gamma }=({\bar{Y}}_{+}+{\bar{Y}}_{-}-2{\bar{Y}}_0)/2$)

Fit a quadratic mean at three temperatures $x_0,x_0\pm 1$, with equal replication $n$ at the off-center temperatures and any positive replication $m$ at the center. The three parameters fit the three group means exactly, giving the displayed estimates. Independent Gaussian errors make these estimates jointly normal. Their covariance is $\frac{1}{4}[\mathrm{Var}({\bar{Y}}_{+})-\mathrm{Var}({\bar{Y}}_{-})]=0$, so they are independent. Equal off-center replication is essential to this cancellation.