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A mean square in ANOVA divides a sum of squares in ANOVA by its positive number $d$ of statistical degrees of freedom. If the corresponding residual subspace has scalar error covariance $\lambda I$, its expectation is $\lambda$. Its scale depends on whether raw observations or group means were projected.

Both factors are assigned at orchard level. Therefore the experimental units are the twelve orchards; the trees are observational units within them. The six combinations form a balanced factorial design, replicated twice. The orchard ANOVA stratum has $12-1=11$ statistical degrees of freedom. Spray uses $2-1=1$, pruning uses $3-1=2$, and their interaction term uses $(2-1)(3-1)=2$, leaving six for error.

Dividing each treatment sum of squares in ANOVA by its statistical degrees of freedom and using $1440/6=240$ as the denominator gives all missing entries:

The unrounded F-test statistics are $998/240=4.15833\dots$, $560/240=2.33333\dots$ and $202/240=0.841667\dots$. The within-orchard tree mean square in ANOVA, 180, is not the treatment error denominator: using it would confuse subsampling with independent replication. The tree ANOVA stratum has $12(30-1)=348$ statistical degrees of freedom; $1+11+348=360$ is the uncorrected total, and the corrected total is 359.

For an orthogonal projection matrix $P$, a sum of squares in ANOVA is $Y^{T}PY=\Vert PY{\Vert }^2$. Mutually orthogonal projections yield an additive decomposition of total squared response magnitude. Removing the grand mean gives the corrected total.