Error, analytical distribution

The function of the analyst is to obtain a result as near to the true value as possible by the correct application of the analytical procedure employed. The level of confidence that the analyst may enjoy in his results will be very small unless he has knowledge of the accuracy and precision of the method used as well as being aware of the sources of error which may be introduced. Quantitative analysis is not simply a case of taking a sample, carrying out a single determination and then claiming that the value obtained is irrefutable. It also requires a sound knowledge of the chemistry involved, of the possibilities of interferences from other ions, elements and compounds as well as of the statistical distribution of values. The purpose of this chapter is to explain some of the terms employed and to outline the statistical procedures which may be applied to the analytical results. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

For analytical applications it is important to realize that three distributions are involved, namely one that describes the measurement process, one that brings in the sampling error, and another that characterizes the sam-... 

Option (Valid) presents a graph of relative standard deviation (c.o.v.) versus concentration, with the relative residuals superimposed. This gives a clear overview of the performance to be expected from a linear calibration Signal = A + B Concentration, both in terms of (relative) precision and of accuracy, because only a well-behaved analytical method will show most of the residuals to be inside a narrow trumpet -like curve this trumpet is wide at low concentrations and should narrow down to c.o.v. = 5% and rel. CL = 10%, or thereabouts, at medium to high concentrations. Residuals that are not randomly distributed about the horizontal axis point either to the presence of outliers, nonlinearity, or errors in the preparation of standards. 

If the variability (a) depends on concentration, prior knowledge of concentration may be required to use these formulas. If the relative standard deviation (RSD) is constant with respect to concentration, then the formulas can be applied by interpreting a and E as relative standard deviation and relative error, respectively. A common case in which RSD is constant with respect to concentration is when analytical results are lognormally distributed. For example, suppose it is desirable to estimate the average concentration with 95% confidence that the estimate will be within 10% of the true value if the relative standard deviation is 25%. Then... 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

As probabilistic exposure and risk assessment methods are developed and become more frequently used for environmental fate and effects assessment, OPP increasingly needs distributions of environmental fate values rather than single point estimates, and quantitation of error and uncertainty in measurements. Probabilistic models currently being developed by the OPP require distributions of environmental fate and effects parameters either by measurement, extrapolation or a combination of the two. The models predictions will allow regulators to base decisions on the likelihood and magnitude of exposure and effects for a range of conditions which vary both spatially and temporally, rather than in a specific environment under static conditions. This increased need for basic data on environmental fate may increase data collection and drive development of less costly and more precise analytical methods. 

Second, the probability that the assigned analytical errors would yield at least the observed amount of scatter (usually referred to as the probability of fif ) can be calculated from the chi-squared distribution of v x MSWD about v. For example. 

Methods. Perhaps the best way of dealing with this thorny problem (common to not only °Th/U geochronology, but also the more classical methods of isotope geochronology as well) is to abandon the reliance on a strictly Gaussian distribution of residuals, whether arising from analytical error or geologic complexities. Robusf in the statistical sense implies insensitivity to departure of the data from the initial... 

Figure 5.25. (A) Quantitative Cu map of an Al-4wt% Cu film at 230 kX, 128 x 128 pixels, probe size 2.7nm, probe current 1.9 nA, dwell time 120 msec per pixel, frame time 0.75 hr. Composition range is shown on the intensity scale (Reproduced with permission by Carpenter et al. 1999). (B) Line profile extracted from the edge-on boundary marked in Figure 5.25a, averaged over 20 pixels ( 55 nm) parallel to the boundary, showing an analytical resolution of 8nm FWTM. Error bars represent 95% confidence, and solid curve is a Gaussian distribution fitted to the data (Reproduced with permission by Carpenter...

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

The several variants deriving from the items 1 to 4 are represented in the flow sheet given in Fig. 6.6. Common calibration by Gaussian least squares estimation (OLS) can only be applied if the measured values are independent and normal-distributed, free from outliers and leverage points and are characterized by homoscedastic errors. Additionally, the error of the values in the analytical quantity x (measurand) must be negligible compared with the errors of the measured values y. 

As can be seen from the distribution function B in Fig. 7.8, an analytical value Xacv produces only in 50% of all cases signals y > yc. Whereas the error of the first kind (classifying a blank erroneously as real measurement value) by the choice of k = 2... 3 can be aimed at a 0.05, the error of the second kind (classifying a real measured value erroneously as blank) amounts /) 0.5. Therefore, this analytical value -which sometimes, promoted by the early publications of Kaiser [1965, 1966], plays a certain role in analytical detection - do not have any significance as a reporting limit in case of y < yc, when no relevant signal have been found. For this purpose, the limit of detection, Xio, has to be used. 

If error is random and follows probabilistic (normally distributed) variance phenomena, we must be able to make additional measurements to reduce the measurement noise or variability. This is certainly true in the real world to some extent. Most of us having some basic statistical training will recall the concept of calculating the number of measurements required to establish a mean value (or analytical result) with a prescribed accuracy. For this calculation one would designate the allowable error (e), and a probability (or risk) that a measured value (m) would be different by an amount (d). 

---