statistical mechanics - Should annealed disorder be characterized by the average of the partition function?

Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken:

$$\begin{equation} \langle \log Z \rangle, \end{equation}$$

while for the annealed average, it's

$$\begin{equation} \langle Z \rangle. \end{equation}$$

But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead.

Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?