This will be an attempt to use the classical “German Tank Problem” (Wikipedia Link) equations to estimate Starfleet's size through NCC hull numbers.

The Original Series (2260-2295~ Era)

Using Memory Alpha and other sites (Ex Astris Scientia: Starships in TOS + TOS Remastered), plus close ups of a chart from Star Trek VI (Operation Retrieve Prop Closeup, Prop 2 Closeup, Prop 2 Reconstruction), I've constructed a list of ships in service in the 30 year era from TOS to Star Trek VI.

Doing the numbers in Excel, we get the following statistics which are needed for the German Tank Problem:

Maximum Number: 2544
Minimum Number: 602
Number Spread (MaxNum – MinNum): 1942
Sample Size: 34

The simplified “German Tank Problem” Equations are:

Frequentist Prediction (N)

n = NumMax + (NumMax / SampleSize) - 1

Bayesian Median (N_Med)

N_Med = NumMax + (NumMax * ln(2) / SampleSize – 1)

Bayesian Mean (N_av)

N_av = (NumMax – 1) * (SampleSize – 1 / Sample Size – 2)

Where n, N_med, and N_av is the maximum population from the sample provided.

Using the equation(s) above and rounded down to the nearest whole digit, we get:

n = 2617~
nMed =2597
nAv = 2622

Average of all three Ns: 2612

We can estimate, assuming serial numbers are assigned sequentially, that there are about 2600+ starships in service with Starfleet in the TOS era.

The Next Generation (2364-2375 Era)

Using common Star Trek sites, I assembled the following sample list:

Running the equations, we get:

Maximum Number: 75633
Minimum Number: 1887
Number Spread (MaxNum – MinNum): 73746
Sample Size: 31
n = 78071~
nMed =77380~
nAv = 78240~

Average of all three Ns: 77897

As a mental exercise, I increased the Sample Size artificially from 31 to as high as 1000; and n decreased only slightly to 75,707.

We thus estimate, assuming serial numbers are assigned sequentially, that there are about 77,900+ vessels in service with Starfleet in the TNG era.

Over Seventy Thousand? That can't be right!

I know, this stands out for me. I can foresee two possibilities:

• At some point between 2295 and 2364 (some 69 years), the Federation nationalized commerce between worlds – while you had “free traders” with their own ships like Harcourt Fenton Mudd in TOS, by TNG era, you had Federation ships doing routine commercial/cargo services between worlds; and as a result NCC registry numbers were assigned to large numbers of small ships or craft, rather than being reserved for warships.

• The NCC Registry was redesigned between TOS and TNG to be more in line with the TNG era Stardate system; which was described in the 1987 TNG Series Bible as the first two digits always being 41, as 4 stood for the 24th century, and 1 for the first season.

If we go with the NCC registry redesign; looking at Memory Alpha gets these numbers:

2250s (Constitution Class): 956, 1017, 1631, 1657, 1664, 1672, 1700, 1701, 1703, 1709, 1764, 1856, 1895, 2014, 2048
2260s (Miranda Class): 1837, 1864, 1887, 1948, 9844, 21166, 21382, 31060, 31905, 31910, 31911, 32591
2260s (Oberth Class): 602, 638, 640, 18834, 19002, 31600, 53847, 53911, 59318
2280s (Excelsior Class): 2000, 2544, 14232, 14598, 42111, 42285, 42296, 42768, 43305, 62043
2340s (Ambassador Class): 26510, 26517, 26136
2360s (Galaxy Class): 70637, 71099, 71807, 71832, 71854
2360s (Nebula Class): 30405, 60205, 60597, 61826, 61827, 62048, 65420, 70352, 70915, 71201, 72015
2370s (Sovereign Class): 72504, 73811, 74107, 74181, 74669, 74877, 74957, 74975, 75306, 75307
2370s (Intrepid Class): 74656 ,74705

I believe we're seeing the existence of perhaps three different hull classification schemes here:

1.) The First Starfleet Scheme – up to maybe 2240s – Ships are given hull numbers sequentially as they are ordered or commissioned.

2.) The Second Starfleet Scheme – 2240s to 2280s – Ships get given a hull number depending on their size – small ships like the Oberth get three digit numbers, while large ships capable of extended independent operation (Constitution, Miranda, etc) get given four digit numbers.

3.) The Third Starfleet Scheme (TNG/DS9/VOY) – At some point, the second scheme broke down as Starfleet increased in size; forcing four digit hull numbers onto small ships (perhaps in the 2300s and it was decided to redesign the registry – older hull numbers in existence would be preserved for record keeping, but all ships ordered past a certain stardate would be given numbers in accordance with the new scheme; which would use a standardized five-digit system, with the first digit being the construction yard ordered from; giving each major construction yard in the Federation a 9999 vessel slot. Assuming five ships are being worked on at any one time with a three year completion period; that makes the system good for about the next 600 years.