Variance of a treatment contrast ($\mathrm{Var}(c^T\hat{\mu })=c^T\mathrm{Cov}(\hat{\mu })c$)

For a vector of estimated treatment means $\hat{\mu }$, the variance of a treatment contrast with coefficient vector $c$ is $c^T\mathrm{Cov}(\hat{\mu })c$. Averaging subsamples from a common experimental unit does not make their shared variance component disappear.