I think that some of the (rather large amount) of reading you've been pointed to explains this. But to repeat, it is actually measured. A sample is monitored, in some cases for years.
I don't know enough to be able to comment on calculations also being able to predict anything.
From good ol' talk origins
TalkOrigins Archive - Feedback for January 2003
Decay rates are measured in the lab, not merely "assumed." For example, in 1955, Kovarik and Adams reported an experimental determination of the U238 half-life, of 4.507 billion years. Their paper describes one experimental setup:
Alpha particles from a thin layer of natural uranium in the form of oxide U3O8 were allowed to pass through a grid of known geometry into an ionization chamber. The electrical impulses there produced were amplified and applied to an electromechanical recording system. By this means the specific alpha activity of natural uranium was observed to be 1486 disintegrations per minute per milligram.
[Kovarik, A.F., and N.I. Adams, Jr., 1955. "Redetermination of the Disintegration Constant of U238" in Physical Review 98, No. 1, p. 46]
The half-life can be computed directly from the measured rate of disintegration. It doesn't actually require waiting around for half of the atoms to decay, as some might think. The uncertainty in the measurement is related to the number of decays counted, so scientists compensate for slow-decaying isotopes by measuring the decays among a large number of atoms. (Even one milligram of U238 contains over 2,000,000,000,000,000,000 atoms.)