The distribution of repeated measurements

Although the standard deviation gives a measure of the spread of a set of results about the mean value, it does not indicate the shape of the distribution. To illustrate this we need a large number of measurements such as those in Table 2.1. This gives the results of 50 replicate determinations of the nitrate ion concentration in a particular water specimen, given to two significant figures. 

This set of 50 measurements is a sample from the very large (in theory infinite) number of measurements which we could make of the nitrate ion concentration. The set of all possible measurements is called the population. If there are no systematic errors, then the mean of this population, denoted by p, is the true value of the nitrate ion concentration which we are trying to determine. The mean of the sample gives us an estimate of ju. Similarly, the population has a standard deviation, denoted by a. The value of the standard deviation, s, of the sample gives us an estimate of a. Use of equation (2.2) gives us an unbiased estimate of a. If n, rather than (n - 1), is used in the denominator of the equation, the value of s obtained tends to underestimate a (see p.l9). 

Tabie 2.2 Frequency table for measurements of nitrate ion concentration 

The measurements of nitrate ion concentration given in Table 2.2 have only certain discrete values, because of the limitations of the method of measurement. In theory a concentration could take any value, so a continuous curve is needed to describe the form of the population from which the sample was taken. The mathematical model usually used is the normal or Gaussian distribution which is described by the equation 

For a normal distribution with mean pi and standard deviation cr, approximately 68% of the population values lie within l rof the mean, approximately 95% of the population values lie within 2 rof the mean, and approximately 99.7% of the population values lie within 3cr of the mean. 

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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.

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.

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. 

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... 

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. 

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. 

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.)... 

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

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)... 

Precision is the closeness of agreement between independent test results obtained under stipulated conditions. Precision depends only on the distribution of random errors and does not relate to the true value. It is calculated by determining the standard deviation of the test results from repeat measurements. In numerical terms, a large number for the precision indicates that the results are scattered, i.e. the precision is poor. Quantitative measures of precision depend critically on the stipulated conditions. Repeatability and reproducibility are the two extreme conditions. 

FIGURE 4.32 Comparison of the measured y-values of the PAC data set with the predictions using PCR with 21 components. The 100 predictions to each y-value from repeated double CV (right) give a better insight into the distribution of the prediction errors than the results from a single CV (left). 

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. 

When one considers the distribution of trace components in dynamic systems, it must also be accepted that the distribution will not necessarily be uniform either in space or time. Nevertheless, most people are attuned to a sense that analytical measurements should be highly repeatable. In consequence, there is far too often a strong tendency to discount or eliminate from consideration, measurements in a sample series which deviate... 

Precision determines the reproducibility or repeatability of the analytical data. It measures how closely multiple analysis of a given sample agree with each other. If a sample is repeatedly analyzed under identical conditions, the results of each measurement, x, may vary from each other due to experimental error or causes beyond control. These results will be distributed randomly about a mean value which is the arithmetic average of all measurements. If the frequency is plotted against the results of each measurement, a bell-shaped curve known as normal distribution curve or gaussian curve, as shown below, will be obtained (Figure 1.2.1). (In many highly dirty environmental samples, the results of multiple analysis may show skewed distribution and not normal distribution.)... 

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