Dear Friends:
I have not seen an answer to the question on how to estimate bee
populations. (Please forgive me if I am not interpreting this question
correctly). There is a simple method for this procedure. I hope that I
can explain it simply enough for everyone to understand.  I learned to
do the estimate using the metric system, hence it is easy for me to
explain it using the metric system.  I think that once you see my
explanation, those of you who wish may convert to inches if desired.
(I'll use Langstroth frame measurements).
      Langstroth frames measure 42 x 20 cms or 840 square centimeters.
Add both sides and we have 1,680 square centimeters. ( Given:  each ten
square centimeters have 425 worker brood cells).  Hence          160
square centimeters
                                                x   425 cells per square
centimeter
                                                =   6,800 potential
worker brood cells
Potential because as we all know, only about two thirds of any frame
contains brood cells, the balance going to honey storage or remaining
empty.
Again doing math calculations a la 1930 style:
                             6,800 potential brood cells
                             67.33 (two thirds of the cells utilized for
worker brood cells)
                        =  4,518 brood cells
    Assuming that we are using 10 frame brood chambers, we can estimate
our brood to be 45,000 strong.  Assuming that all of these mature to
adults, our worker population can
be safely estimated at 45,000 strong.  You can make your own
calculations (rough estimates about the drone population if you care.
Just remember that drones serve one purpose only (besides playing host
to Varroa mites in modern times) and that there are
a lot of them around during mating time, but calculating their numbers
to me have little economic value, (except that they take up worker brood
cells), thus I would not bother.
I hope that I have made my explanation understandable by all readers.  I
can give you estimates for other types of frames, but I figure that you
can do that by merely substituting figures (dimensions of the frames).
Incidentally, I learned to estimate bee populations from one of the
World's great beekeepers, Professor,  Dr. Andres Romero Fabrega, Leon,
Spain: Explotacion
Racional del Colmenar, Editorial Sintes, Barcelona, Spain, (1974).
Best regards.
Dr. Rodriguez
Virginia Beach, VA