Why is the F-measure preferred for classification tasks? Why is the F-measure usually used for (supervised) classification tasks, whereas the G-measure (or Fowlkes–Mallows index) is  generally used for (unsupervised) clustering tasks?

The F-measure is the harmonic mean of the precision and recall.

The G-measure (or Fowlkes–Mallows index) is the geometric mean of the precision and recall.

Below is a plot of the different means.



F1 (harmonic) $= 2\cdot\frac{precision\cdot recall}{precision + recall}$

Geometric $= \sqrt{precision\cdot recall}$

Arithmetic $= \frac{precision + recall}{2}$

The reason I ask is that I need to decide which average to use in a NLG task, where I measured BLEU and ROUGE ( where BLEU is equivalent to precision and ROUGE to recall). How should I calculate the mean of these scores?
The Fı-score is preferred to simple classification accuracy in order to counter the problem of imbalanced datasets; if the thing you are looking for occurs only rarely anyway then a naive classifier can always say no and appear to be working very well! A variant on Fı is Fß, where 

Fß = (1+ß²) &times; [ (P &times; R) &divide; ( (ß² &times; P) + R ) ]

Vary ß to balance precision and recall. As to the why F or G, I believe it to be empirical - you don't say if you are classifying or clustering in your own application?