1994 Volume 43 Issue 6 Pages 325-333
A suitable calculation method to get the averaged values of the radioactivities scattered in wide regions is discussed.
A preliminary analytical calculation in which no background exists is carried out before a computer simulation is done. In this case, an exact maximum-likelihood solution can be obtained, if only the distribution of the parent mean of the activities is assumed to have a gamma-distribution.
In this calculation, the simplest arithmetic mean suggests to give good estimates to the parent mean.
In the case that the backgrounds are to be considered, no exact maximum-likelihood solution can be obtained, and the computer simulation method can only be applied to compare the methods.
The parent mean of the activities is assumed to be beta-distributed by the reason of the easiness of the treatment of the functional forms.
The computer simulations are carried out to get two averaging methods, the simplest arithmetic mean and conventional weighted mean used for Gaussian-distributed data, the accumulation times being decided by the random numbers.
Three to ten measured data in the region are averaged according to the methods, and the same procedures are repeated 10 000 times with the different random number series.
In the results, the simplest arithmetic means always give good results for estimating the parent mean. The conventional weighted means always give not good estimates and give the smaller values than the parent mean.
An exact method to get a sample standard distribution for Poisson-distributed data cannot be applied in the present case, because the parent mean distributes and has no constant value.
In the case that the parent mean does not distribute, a comparison of the methods to get standard deviations is carried out for the exact method and the conventional one for Gaussian-data. The former gives always close values to the theoretical parent standard deviations and the latter gives strong discrepancies from the parent values, if the data numbers are small. The ratios of the both values do not depend on the parent means, in spite of their strong dependences on the data numbers. This suggests that the ratios can be used as the correction factors to get good estimates of the errors which are to be reported.
The standard deviation obtained by the conventional method in the computer simulation are multiplied by the correction factors, and the good results can be obtained.
