## Counting Statistics

The use of statistical methods is important in counting radioactive decay events. We illustrate this with radioactive dating (see also Appendix K). For the decay of carbon-14, the presently accepted values of the radioactive decay constants are

Half-life: T1/2 = 5730 years, Lifetime: t = 8268 years, where T1/2 = 0.693 t, as described in equation (G.9). Thus the number of carbon-14 atoms decreases by 1% in every 83 years, indicating that if you can determine the amount of radioactive carbon-14 with sufficient accuracy, then you can determine a time scale.

The time scale is based on the fact that organic matter, while it is alive, is in equilibrium with the cosmic radiation, and the radiocarbon atoms that disintegrate in living things are replaced by the carbon-14 entering the food and growth chain. According to the law of radioactive decay, after 5730 years the carbon that was in the body while it was alive will show half the specific carbon-14 radioactivity that it showed at the time of death. In the disintegration process, the carbon-14 returns to nitrogen-14 by emitting a beta particle.

In radioactive dating one often must detect small differences in the amounts of carbon-14 present. The difference between a present sample and one 83 years old (one-hundredth of a lifetime) is only 1% of the amount of carbon-14 present in the two samples. Counting statistics are an important issue. If N is the number of particles counted, then there is a 68% probability that the true value (corresponding to the average rate multiplied by the counting time) lies within the limits where N ± a, where a, the standard error, is equal to N1/2. See Table G.1.

If the error limits are widened to ±2a, then there is a 95% probability that the true value is contained within them, and for ±3a the probability is 99.7%. This statistical uncertaintly implies uncertainty about the true value of the radiocarbon age.

 TABLE G.1 For a = N1/2 the following relations hold: N = 100 a = 10, i.e., 10% of N N = 1000 a = 33, i.e., 3.3% of N N = 10,000 a = 100, i.e., 1% of N N = 100,000 a = 330, i.e., 0.33% of N Counting statistics for N events with standard deviation a for a = n1/2.

Fig. G.3. Probability curve of a normal or Gaussian distribution plotted versus standard error (or deviation) a. The probability is 68% that an event occurs within the cross-hatched area between plus and minus one standard deviation (± 1a). The probability is 95% for ±2<x and 99.7% for ± 3a (Taken from Aitken, Science-Based Dating in Archaeology (Longman, London, 1990)).

The statistical uncertainty is due to the fact that radioactive decay is a random event; that is, the decay of any individual nucleus is a random event in time that we can assign a probability per unit time that the decay occurs. With a million radioactive decays the uncertainty is small—the square root of 106 is 103—but with one hundred events, the standard is 10, equivalent to 10%, or a dating uncertainty of plus or minus about 830 years for C-14 dating. The probability is plotted as a Gaussian distribution in Figure G.3. This statistical uncertainty due to the randomness of radioactive decay is a fundamental limitation in the precision to which beta activity may be measured. There are also uncertainties besides the statistical one.  