Low numbers of counts

If the number of counts accumulated is small, then, even though the count distribution will be Poisson, the approximation to a Normal distribution will not be valid. This means that the relationships to calculate the decision limits given above will not be valid. For number of counts of less than 25, we must resort to the Poisson distribution itself. 

For example, if we wish to calculate the critical limit, Lq, we must consider the distribution of counts when the 

The probability that a blank sample could have a count greater than Lq, for a particular degree of confidence defined by a, is given by  

For small numbers of counts, the Poisson distribution is not symmetrical. This implies that confidence limits associated with a small count will not be symmetrical either. Again, appropriate limits can be tabulated and examples. 

A number of authors have commented on the failure of the Currie equations to cater adequately for the low count situation and proposed alternative equations for critical and detection limits for the single count situation. Strom and MacLellan (2001) compare eight equations for calculating the critical Umit, looking at their tendency to allow false positive identifications. None of the rules appears to be completely satisfactory but the authors advocate the use of a Stapleton rule rather than the usual Currie equation in situations where background counts are very low and a, as in Section 5.6.1, is 0.05 or less. 

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The resulting spectrum of 140 measured in ten independent runs is shown in fig. 5. Despite the rather low number of counts, the enhancement at 53 MeV is clearly observed. We have attempted to fit the spectrum with the five-body 140+3H+ n + n+n phase space contribution (dashed line) but with little success. In order to explain the experimental data which cannot be reproduced by either five- or four-body phase space we had to include the three-body exit channel 140 + 5H + n, though neither bound nor unbound levels have been observed in the 9Be(uB, 150)5H reaction. A similar situation has appeared in the 7Li(ir, ir+)7H reaction12), where the contribution from the 7H- 5H + n + n exit channel had to be included. This fact can only be understood as a consequence of the final state interaction in the SH system with a very large width (T 10 MeV), that makes it very difficult to observe this interaction as a peak in the reactions leading directly to the 5H nucleus. 

Although radioactive decay is random, the data, particularly for low number of counts, are not distributed according to Equation al-14 (Appendix 1), because the decay process does not follow Gaussian he-havior. The reason that decay data arc not normally distributed is that radioactivity consists of a series of discrete events that cannot vary continuously as can the indeterminate errors for which the Gaussian distribution applies. Furthermore, negative numbers of counts are not possible, rhcrcforc, the data cannot be distributed symmeirically about the mean. 

Thus, the various simulated tissues can be identified using the above-described frequency analysis. Fat and bone marrow had a low number of counts at the generated frequencies. Spongy bone had different peak frequencies from the other samples. Meniscus and cortical bone had few counts above a frequency of 50 kHz, although pork, meniscus, and cortical bone have similar peak frequencies. This indicates that tissue can be identified using the above-described frequency analysis. 

Equation 10-6 is the well-known Poisson distribution,5 which is important in counting whenever the number of counts taken is low enough to make a count of zero fairly probable. The analytical chemist, except occasionally in trace determinations, wrill deal with counts so large that he need not concern himself with the Poisson distribution. 

There are some problems associated with this experiment in terms of low statistical precision, uncertainity in S02 concentrations, and the presence of butanol in the system. Statistical precision is low because of the small number of counts and low number of data points taken. Thus, the multiple peaks observed in Figure 3 may actually be only 2 distinct peaks where the broad peak has mobilities ranging from 0.4 cm2V 1s to 1.6 cnrV s and the narrow peak has mobilities centered around 2.0 qwlV s. These peaks thus correspond to those reported by Bricard et al. (1966). 

By this technique, the degree of association can be pictorially presented. The number of counts in each box will depend on the cutoff points between good and poor stability, and between the low and high levels of contaminants. These can be varied to investigate the effect of changing special limits and levels of control. 

The selection of an appropriate technique for low level counting has been discussed by Moghissi et al. (9). If M ml. of water with a specific activity of Y nCi/liter are counted with an efficiency of E c.p.m./d.p.m. for t minutes, the number of counts obtained are as follows ... 

The boundaries defining the central part of the pulser peak are set to the positions where the ratio between the contents of two successive channels, Ni+1/N, reaches its maximum on the low-energy side and its minimum on the high-energy side of the peak. The number of counts in the pulser peak is calculated by summing up the counts in its central region and the counts registered in its tails nTL and nTH ... 

Count the number of colonies present on the IPTG/Xgal/amp plate containing bacteria from transformation culture tube 2. Why was this control experiment performed What color are these colonies Explain. Describe what a low number of colonies on this plate would indicate, as well as possible causes for this result. 

These apply to discrete characteristics which can assume low whole-number values, such as counts of events occurring in area, volume or time. The events should be rare in that the mean number observed should be a small proportion of the total that could possibly be found. Also, finding one count should not influence the probability of finding another. The shape of Poisson distributions is described by only one parameter, the mean number of events observed, and has the special characteristic that the variance is equal to the mean. The shape has a pronounced positive skewness at low mean counts, but becomes more and more symmetrical as the mean number of counts increases (Fig. 41.3). 

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