= Independent slope and curvature estimates in a symmetric quadratic design
{title2=$\widehat\beta=(\bar Y_+-\bar Y_-)/2,\quad\widehat\gamma=(\bar Y_++\bar Y_--2\bar Y_0)/2$}
Fit a quadratic mean at three temperatures $x_0,x_0\pm1$, 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 $\tfrac14[\operatorname{Var}(\bar Y_+)-\operatorname{Var}(\bar Y_-)]=0$, so they are independent. Equal off-center replication is essential to this cancellation.
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