How was the half life of carbon 14 determined?
Experimentally and by extrapolation
The level of radioactive Carbon was measured in organisms that had just died. This is a very small value and multiple experiments were done to measure the levels and the results averaged.
The level of radioactive Carbon is assumed to equal the level of radioactive Carbon in the atmosphere which is assumed to have remained constant through out the history of the earth. ( An assumption known to be wrong but necessary for the calculations)
The samples were then measured after a period of time to find the new levels of radioactivity. The loss of radioactivity was then extrapolated backwards to find the half life.
The results were then tested and refined by measuring the levels of radioactive carbon in samples of organic material from archeological sites of known ages. The levels of radioactive material in a house built in 1200 AD could be determined and compared to the extrapolated experimental curve. The assumption that the wood used in the house was close to the known age of the construction of the house leaves a plus or minus value to the age determination of the wood.
Multiple checks of known ages to the experimental curve have been done resulting in the present estimated half life of radioactive Carbon. This value has been revised from previous estimates resulting in a more precise value for the half life.
Willard F. Libby, at the University of Chicago, developed the process of radioactive carbon dating in 1948.
A couple of very good historical reviews and details can be found here: