On the F1 score sklearn page there's a section that explains each of the options for the average parameter. Under the weighted option, it says: "it can result in an F-score that is not between precision and recall."
I would like to know why this happens. Thanks
It appears this can happen already with the macro average option. The statement needs some clarification, but I assume the precision and recall that are supposed to not bound the averaged F1 are themselves the same type of average.
Here's a simple example: $TP=TN=4$, $FP=1$, $FN=16$. Then $$\begin{align*} \operatorname{precision}(1)&=\frac{TP}{TP+FP}=0.8, \\ \operatorname{recall}(1)&=\frac{TP}{TP+FN}=0.2, \\ \operatorname{precision}(0)&=\frac{TN}{TN+FN}=0.2, \\ \operatorname{recall}(0)&=\frac{TN}{TN+FP}=0.8 \end{align*}$$
and so $F_1(1)=F_1(0)=0.32$, so the macro-average $F_1$ is also $0.32$. But the macro-averaged precision and recall are both $0.5$.
the F1 score uses a harmonic mean rather than the actual mean, which accounts for the difference
