On the calculation of the average in radioactivity measurements

Three different methods for averaging the independent experimental results in radioactivity measurements are compared by using computer simulation. The first method is newly introduced according to the maximum likelihood method. The second and third methods are the conventional simplest arithmetic and weighted means, respectively. The parent mean of the net counts per unit time, μ, is assumed to be identical in two measurements and the parent means of the background per unit time, λ1 and λ2, are different each other. The counting data of the sample χκ (κ=1, 2), are generated by the Poisson random number generator, whose parent means are of (μ+λκ)tκ, respectively, where tκ are the accumulation periods of the sample measurements. The background data are also generated with the parent means, λκτκ (κ=1, 2), where τκ are the accumulation periods of the background. The generated data are analysed by the different three methods. In the new method, one can get non-linear simultaneous equations by differentiating the likelihood function with respect to μ, λ1 and λ2. The simultaneous equation can be solved by a successive approximation, and then the estimate of μ can be obtained. The simplest arithmetic mean is carried out for the sample counts per unit time subtracted by the backgrounds per unit time. The weighted mean is done in the same manner as the case of the Gaussian-distributed data. These processes are repeated 10,000 or 20,000 times and the arithmetic means of three estimates are independently calculated according to the methods. The new method and the simplest arithmetic mean denote the similar results, and give good estimates of the parent means, μ. The weighted mean does not give good estimates. The new method is most desirable for the analysis, if all of the original data in measurement are known. (J.P.N.)