Statistics of counting

The collection of data in Table 3-11 can be used to provide an illustration of the basic terms used in statistics. In the first column of figures are the observed estimates, x for a sample of radioactivity. The average or mean value of these estimates, x is 26,644 and is a better estimate of the true amount of radioactivity in the sample than any of the observed data. The mean value approaches the true amount of radioactivity in the sample as the number of independent estimates approaches infinity. A measure of the scatter observed in the 10 estimates is shown in the second column. These values, termed the deviations, are obtained by subtracting the mean from each of the estimates. 

Notice that the number of times the estimate is greater than the mean is 

X is not usually available because most experimental samples are counted only once. As a result of this limitation it has become customary to indicate a proportional error of a given count. At a confidence level of 95% (i.e., there is a 95% chance the observed value will fall within the interval x - l-96o-) the proportional error may be calculated from the relationship 

When the first RNA polymerase reaches the end of the operon it drops off and releases a free RNA chain, which for the moment we inaccurately assume has not undergone any degradation. 

A pulse-chase experiment is merely a variation of the pulse labeling 

In this chapter, I will examine the statistical nature of radioactivity counting. Statistics is unavoidably mathematical in nature and many equations will emerge from the discussion. However, only as much general statistical mathematics wiU be introduced as is necessary to understand the relevant matters. I wiU go on to discuss the statistical aspects of peak area measurement, background subtraction, choosing optimum counting parameters and the often superficially understood critical limits and minimum detectable activity. I end with an examination of some special counting situations. 

At its simplest, radioactivity counting involves a source, a suitable detector for the radiation emitted by the source, a means of counting those decay events that are detected and a timer. If we measure the rate of detection of events, we can directly relate this to the number of radioactive atoms present in the source. The basic premise is that the decay rate of the source (/ ) is proportional to the number of atoms of radioactive nuchde present (N), the proportionality constant being the decay constant, X. Thus  

of course, what would normally be referred to as the activity of the sample. In principle, therefore, if we count the number of events, C, detected by the detector in a fixed period of time. At, we can estimate the decay rate as follows  

While it is true to say that all scientific measurements are estimates of some unattainable true measurement, this is particularly true of radioactivity measurements because of the statistical nature of radioactive decay. Consider a collection of unstable atoms. We can be certain that all wiU eventually decay. We can expect that at any point in time the rate of decay will be that given by Equation (5.1). However, if we take any particular atom we can never know exactly when it will decay. It follows that we can never know exactly how many atoms will decay within our measurement period. Our measurement can, therefore, only be an estimate of the expected decay rate. If we were to make further measurements, these would provide more, slightly different, estimates. This fundamental uncertainty in the quantity we wish to measure, the decay rate, underlies ah radioactivity measurements and is in addition to the usual uncertainties (random and systematic) imposed by the measurement process itself. 

At this point, it is appropriate to introduce a number of statistical relationships with which I can describe the distribution of a number of measurements. This section mnst necessarily be somewhat mathematical. However, textbooks on statistics will cover the theoretical basis of these parameters in much detail, and here I will content myself with a number of simple definitive statements. Later, these will become relevant to an understanding of counting statistics. 

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In NAA it is possible to study separately diflFerent sources of error and to come up with a good estimate of how they affect the accuracy of determination for each element. A basic source of error is introduced by the statistics of counting radioactive emissions and is called the counting error. The size of the error is calculated directly from all of the counts used in the analysis of each peak. These errors vary considerably from element to element, but for our purposes we are only interested in those peaks that have counting errors of a few percent or less. 

A large variance for Na by INAA could be explained by the application of a low neutron flux and short irradiation and measuring times the uncertainty was related to the statistics of counting low numbers. A large standard deviation was also observed for Fe, also dictated by counting statistics. 

The need for high counting rates arises from the analytical error inherent in the statistics of counting random events. In fluorescence analyzers having a constant excitation intensity, the detection of x-ray photons constitutes the counting of random events and obeys the laws of Poisson statistics. Crudely speaking, the... 

The statistics of counting in the case of low level radionuclide activity. 

In general, we assume that the statistics of counting can be adequately described by the Poisson distribution. When we calculated the various decision limits, we effectively assumed, for simplicity, the Normal distribution for the counts. We know, however, that Poisson statistics are only applicable when the probability of detection of the decay of any particular radioactive atom within the count period is small and when the statistical sample size is large. There are a number of circumstances when these conditions may not be met and we should consider whether the statistical treatment above is still valid. 

Coulter has discussed the effects of experimental errors on the precision of i.d.a. The most important factor is the extent of dilution, this having a larger effect than errors due to the statistics of counting, and should be carefully controlled. A nomogram is presented which enables optimum experimental parameters to be predicted for a particular analytical system, or allows an estimation of the precision of an actual system. Klas ° has more recently considered the optimization of experimental factors in i.d.a. with respect to the overall time of analysis. 

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