I have done multiple searches trying to determine the directivity of a phased array antenna. I do realize that a single number for directivity is not applicable to all phased arrays since you can have different scan angles and use an arbitrary amount of elements in an array. In light of this, I won't ask for an empirical, single, numerical answer, but rather I'd like to determine a general intuition for phased array antenna directivity .
Assume one has chosen correct phasing weights for each element in a phased antenna array to achieve maximum directivity at some angular coordinate $$(\theta,\phi)$$ Also, the total radiated power of each element is given by $$P/n$$ where
P: Sum of power radiated by all elements and
n: number of elements
Is said array's directivity better or worse, at that same angular coordinate, than using a single element with $$P$$ as the total radiated power? $$\ $$
NOTE(1): If it is simpler, assume each element is a perfect isotropic radiator.
A different question that I could ask which would also satisfy my post is the following:
Do phased array antennas offer the advantage of creating a very selective, narrow, main lob that can be steered alone or do they also allow for a greater directivity compared to an isotropic radiator with equal total radiated power?