The process of radiation counting is well-defined by a Poisson distribution for which the mean is equal to the number of counts over a given time period and the standard deviation is equal to the square root of the number of counts [6, 7]. As mentioned earlier, this uncertainty measure quantifies random noise in point measurements for counting processes. The developer of the x-ray absorption apparatus, GNI, states that the Poisson standard deviation is roughly the uncertainty of the measurements. For example, a measurement of 104 counts has an uncertainty of approximately 100 counts or 1 % of the total counts. Likewise, for 1000 counts, the uncertainty is 32 counts or 3 % of the total counts. In other words, fewer counts have a higher uncertainty as a percentage of the counts.
The use of the Poisson approach to estimate uncertainty is only warranted in the x-ray absorption process for one point or for multiple points when the specimen is uniform in composition and thickness. Different points across a heterogeneous specimen have different counts not due to random noise, but to real physical features. For determining the average uncertainty of a vertical scan of a heterogeneous specimen, then, statistical measures other than the Poisson approach would be more appropriate. The variability and the uncertainty of the measurements are also described in this work with the standard deviation (SD) and with the root mean square error (RMSE) and the SD and RMSE normalized by the average number of counts per specimen point for a given measurement (NSD and NRMSE):
where x is the number of counts at a given point, xc is the number of counts at a given point averaged over several samples (the "correct" or "true" value; see below), n is the number of points, and N indicates the average number of counts per point. To compute the RMSE and NRMSE, the average of many measurements was selected as the "correct" or "true" measurement.