Distribution of repeated measurements

Even if all systematic error could be eliminated, the exact value of a chemical or physical quantity still would not be obtained through repeated measurements, due to the presence of random error (Barford, i985). Random error refers to random differences between the measured value and the exact value the magnitude of the random error is a reflection of the precision of the measuring device used in the analysis. Often, random errors are assumed to follow a Gaussian, or normal, distribution, and the precision of a measuring device is characterized by the sample standard deviation of the distribution of repeated measurements made by the device. [By contrast, systematic errors are not subject to any probability distribution law (Velikanov, 1965).] A brief review of the normal distribution is provided below to provide background for a discussion of the quantification of random error. 

Figure 14-7 Outline of basic error model for measurements by a field method. Upper part The distribution of repeated measurements of the same sample, representing a normal distribution around the target value (vertical line) of the sample with a dispersion corresponding to the analytical standard deviation, Oa- Middle part Schematic outline of the dispersion of target value deviations from the respective true values for a population of patient samples, A distribution of an arbitrary form is displayed.The vertical line indicates the mean of the distribution. Lower part The distance from zero to the mean of the target value deviations from the true values represents the mean bias of the method.

Another property of the sampling distribution of the mean is that, even if the original population is not normal, the sampling distribution of the mean tends to the normal distribution as n increases. This result is known as the central limit theorem. This theorem is of great importance because many statistical tests are performed on the mean and assume that it is normally distributed. Since in practice we can assume that distributions of repeated measurements are at least approximately normally distributed, it is reasonable to assume that the means of quite small samples (say >5) are normally distributed. 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Figure 1.9. A large number of repeat measurements x,- are plotted according to the number of observations per x-interval. A bell-shaped distribution can be discerned. The corresponding probability densities PD are plotted as a curve versus the z-value. The probability that an observation is made in the shaded zone is equal to the zone s area relative to the area under the whole curve.

In everyday analytical work it is improbable that a large number of repeat measurements is performed most likely one has to make do with less than 20 replications of any detemunation. No matter which statistical standards are adhered to, such numbers are considered to be small , and hence, the law of large numbers, that is the normal distribution, does not strictly apply. The /-distributions will have to be used the plural derives from the fact that the probability density functions vary systematically with the number of degrees of freedom,/. (Cf. Figs. 1.14 through 1.16.)... 

Random deviations (errors) of repeated measurements manifest themselves as a distribution of the results around the mean of the sample where the variation is randomly distributed to higher and lower values. The expected mean of all the deviations within a measuring series is zero. Random deviations characterize the reliability of measurements and therefore their precision. They are estimated from the results of replicates. If relevant, it is distinguished in repeatability and reproducibility (see Sect. 7.1)... 

Increasing the number of repeated measurements to infinity, while decreasing more and more the width of classes (bars), normally leads to a symmetrical bell-shaped distribution of the measured values, which is called Gaussian or normal distribution. 

We see from the thickness nonuniformity that if the TiN thickness is fixed, the oxide thickness has a lower standard deviation, indicating a tighter distribution of the measurement results. Since the measurement is on the same wafer, the difference suggests that the effect of fixed TiN thickness to help improve the repeatability of the measurement. (Of course, the TiN film must be uniform for this to be true.) In short, if the control of the TiN... 

All sources of uncertainty that are not quantified by the standard deviation of repeated measurements fall in the category of Type components. These were fully dealt with in chapter 6. For method validation, it is important to document the reasoning behind the use of Type components because Type components have the most subjective and arbitrary aspects. Which components are chosen and the rationale behind the inclusion or exclusion of components should documented. The value of the standard uncertainty and the distribution chosen (e.g., uniform, triangular, or normal) should be made available, as should the final method used to combine all sources. 

A probability distribution function for a continuous random variable, denoted by fix), describes how the frequency of repeated measurements is distributed over the range of observed values for the measurement. When considering the probability distribution of a continuous random variable, we can imagine that a set of such measurements will lie within a specific interval. The area under the curve of a graph of a probability distribution for a selected interval gives the probability that a measurement will take on a value in that interval. 

The case studies discussed above, and depicted in Figs. 4.8-4.10, reveal the importance of repeated measurements, providing evolution with time, and the importance of auxiliary data, such as distribution of local precipitation or discharge in adjacent pumping wells. 

As an example, consider the data on day-to-day and interindividual variability of fruit growers respiratory and dermal exposure to captan shown in Table 7.3 (de Cock et al., 1998a). The ratio of the 97.5th percentile to the 2.5th percentile of the exposure distribution R95) is usually larger for the intraindividual or day-to-day variability, when compared to the interindividual variability. The variance ratio, k, can be calculated from the Rgs values, since the standard deviation of each exposure distribution is equal to In R95/3.92, and the square of the standard deviation gives the variance. For the respiratory exposure, this results in a variance ratio k of 32.8, whereas for dermal exposure of the wrist the variance ratio is considerably lower, approximately 3.0. What are the implications of these variance ratios for the number of measurements per study subject For a bias of less than 10 % (or /P > 0.90), the number of repeated measurements per subject... 

Median me-de-9n n. The value in an arrayed set of repeated measurements that divides the set into two equal-numbered groups. If the sample size is odd, the medium is the middle value. The median is a useful measure of the center when the distribution is strongly skewed toward low or high values. Compare arithmetic mean. 

Fig. 29.4 Propoties of weighing insbumcaits The dashed line with the assoeialed grey area represents the sensitivity offset of the instrnment, snpeiimposed is the nonlinearity (blue area, indicating the deviation of the characteristic curve from the straight line). The red circles represent the measurement values caused by eccentric loading, and the yellow circles represent the distribution of the measurement values due to repeatability, from Mettler-Toledo [13] with permission...

The definition of the limit of detection (LOD) has been interpreted to be approximately three times the standard deviation derived from multiple measurements of a blank solution. This assumption is misleading, as it bears several problems. First, with the definition given, the LOD cannot be derived in the absence of known (or assumed) distributions of the concentration of a blank solution (or sample), and also of the solution containing the analyte. By multiplying the standard deviation of the blank solution by k = 3, it is assumed that the distribution of concentrations for the analyte-containing solution is identical with that of the blank. Second, all simplified calculations are based on a normal distribution, whereas a Poisson-like distribution of the concentrations of the blank (at level Iq) is more reasonably assumed. In a normal case, the standard deviation (ct) of at the blank level and at the limit of detection (Iq and Ilod respectively) are only estimated are only estimated (sq is an estimate for ao based on a limited number of blank measurements) by a limited number of repeated measurements and, hence, the number of degrees of freedom (v = u 1) needs to be taken into... 

The distribution of impurities over a flat sihcon surface can be measured by autoradiography or by scanning the surface using any of the methods appropriate for trace impurity detection (see Trace and residue analysis). Depth measurements can be made by combining any of the above measurements with the repeated removal of thin layers of sihcon, either by wet etching, plasma etching, or sputtering. Care must be taken, however, to ensure that the material removal method does not contaminate the sihcon surface. 

Joly observed elevated "Ra activities in deep-sea sediments that he attributed to water column scavenging and removal processes. This hypothesis was later challenged with the hrst seawater °Th measurements (parent of "Ra), and these new results conhrmed that radium was instead actively migrating across the marine sediment-water interface. This seabed source stimulated much activity to use radium as a tracer for ocean circulation. Unfortunately, the utility of Ra as a deep ocean circulation tracer never came to full fruition as biological cycling has been repeatedly shown to have a strong and unpredictable effect on the vertical distribution of this isotope. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

The precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

The measurement of filth elements by microanalysis is a valuable adjunct in the enforcement of the Food, Drug, and Cosmetic Act and serves as an efficient means of evaluating conditions of cleanliness, decency, and sanitation in food-producing plants. This, of course, is in addition to the value of microanalytical methods in the determination of the fitness of foods as they reach the consumer. The techniques available, together with proficiency of manipulation, repeated references to authentic materials, and sound judgment in the interpretation of results, provide effective enforcement weapons in the constant war to prevent the production and interstate distribution of products which are unfit for the table of the American consumer. 

Purpose Determine the distribution of many repeat measurements and compare this distribution to the normal distribution. 

HISTO.dat Section 1.8 19 repeat measurements of a normally distributed... 

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