Relative standard deviation Poisson

The discrete Poisson distribution is only characterized by one parameter, the mean Y. The standard deviation is given by sY = Jy and the relative standard deviation by SyreI = 1 / JY. 

The content uniformity which can be obtained depends, according to established theoretical considerations, on the particle size of the active substance (19,20). As a rough estimate for the obtainable relative standard deviation 5 rei of the content of the active substance, the following rule can be applied, based on Poisson statistics ... 

If a radioactive sample is measured repeatedly with a constant measuring time, the results will scatter, showing a Poisson-type frequency distribution. The precision of a measurement method can be characterized by the relative standard deviation ... 

Precision is a measure of how close repeat measurements of the same sample are to one another. There are two main types of precision, internal and external. When performing a mass spectrometric analysis, several measurements of the elements of interest are made. How close these determinations are to each other defines internal precision. Internal precision generally is a measure of the stability of sample introduction to the mass spectrometer and the stability of the ion source and mass analyzer/detector. Internal precision in mass spectrometry is often counting-statistics limited (see Section 10.3). When the mass spectrometric results are Poisson distributed, the best internal precision that is achievable is N 1/2 where N is the total number of counts. Thus, if 100 counts are registered for some isotope of interest, then this result will only be reproducible, on average, to 10% relative standard deviation. To achieve 0.1% precision, at least 1,000,000... 

Note 5. The relative standard deviation [RSD] of based on observed counts is for Poisson data, or about 6% for -E(N)=60. Equivalent precision for based on replication would require about 2fi or 120 degrees of freedom. The same is true for confidence intervals for n, hence, based on counts, vs based on replication. For more detail, Including the use of x bo derive both types (counts, replication) of Cl s see Ref. and the monograph by Cox and Lewis (100). Adequacy of the large count (normal) approximation, and the exact treatment for extreme low-level data (n 10 or less) are covered in Ref. and the references therein. 

Gray et al. investigated this statement in practice and adapted their quadrupole-based ICP-MS instrument such that as many sources of noise as possible were eliminated, or their contribution minimized [93]. This included the use of a free expansion interface (without skimmer), a bonnet to shield the ICP from the surrounding atmosphere, and the introduction of the analyte (Xe) in gaseous form. Under these extraordinary conditions, they obtained a measurement precision equal to that calculated on the basis of Poisson counting statistics [down to 0.02% relative standard deviation (RSD)]. With a normal (two-cone) interface and sample introduction via pneumatic nebulization, they still obtained extraordinary values of around 0.05%, but at analyte concentrations that are unusually high (100 mg Ag 1 ), an isotope ratio close to 1 ( ° Ag/ ° Ag), and with the sheathing bormet mentioned earlier. 

At low signal levels the noise measured on the ion signal is usually dominated by statistical noise resulting from the random nature of ion arrival at the detector. This statistical noise follows Poisson statistics, which determines that the standard deviation in a set of measurements each containing an average of N ions is The relative standard deviation (RSD) of a set of... 

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

When describing the relative variability of a distribution, however, it is not the ratio of variance to mean which is important but that of standard deviation to mean. This feature is a useful one which can be exploited given an appropriate attitude. The sum of a number of independent Poisson variables is also a Poisson variable, so that if we had four centres each with a mean (Poisson) arrival rate of 9 patients per six months overall, we should have a trial where arrival was described by a Poisson distribution with mean of 36 per six months. Now, since the standard deviation is the square root of the variance, the ratio of standard deviation to mean for a Poisson with mean 36 is 6/36 = 0.17, whereas for a Poisson with mean 9 it is 3/9 = 0.33. 

It is reassuring, however, that in the case of the Poisson distribution the actual difference between the above averages is relatively small. It is easy to check (e.g., with random numbers ofU fi) distribution) that the difference - varying between 10 —10 and averaging ten data at a time - does not usually exceed 1. On the other hand, for = 10, 10, and 10 the standard deviations are 10, 100, and 1,000, respectively, showing that the three methods of averaging are practically equivalent, because the standard deviations are much larger than the difference between them. 

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