Uncertainty counting

For some experiments, the solar neutrino flux and the rate of decay of the proton being extreme examples, tire count rate is so small that observation times of months or even years are required to yield rates of sufficiently small relative uncertainty to be significant. For high count rate experiments, the limitation is the speed with which the electronics can process and record the incoming infomiation. 

The essential quantity is the relative uncertainty in the signal counts, 6 N /N. This is given by... 

The heat capacity can therefore be obtained by keeping a running count of and E during the simulation, from which their expectation values (E ) and (E) can be calculated at the enc of the calculation. Alternatively, if the energies are stored during the simulation then the value of ((E — (E)) ) can be calculated once the simulation has finished. This seconc approach may be more accurate due to round-off errors (E ) and (E) are usually botf large numbers and so there may be a large uncertainty in their difference. 

In research environments where the configuration and activity level of a sample can be made to conform to the desires of the experimenter, it is now possible to measure the energies of many y-rays to 0.01 keV and their emission rates to an uncertainty of about 0.5%. As the measurement conditions vary from the optimum, the uncertainty of the measured value increases. In most cases where the counting rate is high enough to allow collection of sufficient counts in the spectmm, the y-ray energies can stih be deterrnined to about 0.5 keV. If the configuration of the sample is not one for which the detector efficiency has been direcdy measured, however, the uncertainty in the y-ray emission rate may increase to 5 or 10%. 

When measured quantities are added or subtracted, the uncertainty in the result is found in a quite different way than when they are multiplied and divided. It is determined by counting the number of decimal places, that is, the number of digits to the right of the decimal point for each measured quantity. When measured quantities are added or subtracted, the number of decimal places in the result is the same as that in the quantity with the greatest uncertainty and hence the smallest number of decimal places. 

Schwartz, L. M., Statistical Uncertainties of Analyses by Calibration of Counting Measurements, Anal. Chem. 50, 1978, 980-985. 

Unlike non-radiometric methods of analysis, uncertainty modelling in NAA is facilitated by the existence of counting statistics, although in principle an additional source of uncertainty, because this parameter is instantly available from each measurement. If the method is in a state of statistical control, and the counting statistics are small, the major source of variability additional to analytical uncertainty can be attributed to sample inhomogeneity (Becker 1993). In other words, in Equation (2.1) ... 

The denominator in (3.1) can be simplified because the statistical uncertainty of the baseline, hN o, is negligible in practice when the spectra are simulated with numerical line fit routines. The stochastic emission of y-rays by the source leads to a Poisson distribution of counts with the width AA = and since is small, the denominator of (3.1) can be written as ... 

Measuring the isotopic composition of U in estuaries has the potential for further constraining the interpretations of uranium behavior. However, this has been hampered by large uncertainties in conventional methods using counting techniques. While rivers often display ( " U/ U) activity ratios above equilibrium, the ratios generally do not... 

Fig. 13. The total translational energy distributions for the dissociation of 03 to 0(3Pj) + 02(X%-) at 226, 230, 233, 234, 240 and 266nm. The vibrational levels of the 02(X3S ) fragment are indicated by the combs. The dotted curves represent the uncertainty in the signal intensity arising from counting statistics.

Since the counting of radioactive disintegration events is subject to statistical considerations, the dates obtained with this method are inherently subject to some degree of uncertainty. 

Fig. 7.7. Effects of Poisson photon noise on calculated SE and FRET values. (A) Statistical distribution of number of incoming photons for the mean fluorescence intensities of 5,10, 20, 50, and 100 photons/pixel, respectively. For n = 100 (rightmost curve), the SD is 10 thus the relative coefficient of variation (RCV this is SD/mean) is 10 %. In this case, 95% of observations are between 80 and 120. For example, n — 10 the RCY has increased to 33%. (B) To visualize the spread in s.e. caused by the Poisson distribution of pixel intensities that averaged 100 photons for each A, D, and S (right-most curve), s.e. was calculated repeatedly using a Monte Carlo simulation approach. Realistic correction factors were used (a = 0.0023,/ = 0.59, y = 0.15, <5 = 0.0015) that determine 25% FRET efficiency. Note that spread in s.e. based on a population of pixels with RCY = 10 % amounts to RCV = 60 % for these particular settings Other curves for photon counts decreasing as in (A), the uncertainty further grows and an increasing fraction of calculated s.e. values are actually below zero. (C) Spread in Ed values for photon counts as in (A). Note that whereas the value of the mean remains the same, the spread (RCV) increases to several hundred percent. (D) Spread depends not only on photon counts but also on values of the correction...

The variations in the background, the sensitivity to moisture, the alpha activity of the chamber itself and the influence of recombination were discussed by Hultqvist. The standard deviation due to counting statistics was estimated to be about 3 % (in a few measurements 6 %). The calibration was made by counting each alpha particle by a proportional counter specially designed at the Department for this purpose. The statistical uncertainty of the calibration of the equivalent radon concentration was estimated to be 12 %. 

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