Error Analysis of Experiments

Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E(y), the true mean, with variance cTin, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each days measurement, n. Of course, other factors (weather, weekends) may make the observations on different days not distributed identically. 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables yt], Y = oqt/1 + a2y2 + . Let the random variables y2, have means E(yi), E(y2),. . . and variances G2(yi), G2(t/2). The variable Y has mean 

If the variables are uncorrelated and have the same variance, then 

Next suppose the model relating Y to yd is nonlinear, but the errors are small and independent of one another. Then a change in Y is related to changes in yt by 

If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E(dY), is zero. The variances are given by the following equation (Baird, 1995 Box et al., 2005)  

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A common feature of these errors is that they are likely to occur independently of each other. As we shall see, this simplifies the analysis of experiments, since it is often possible to use known statistical distributions to assess the significance of the results obtained, e.g. the normal distribution, or other distributions related to the normal distribution, such as the t distribution or the F distribution. Significance tests based on these distributions are discussed later in this chapter. 

A comparison of our error analysis of Nessler s analytical method for the particular sample types of interest here with the analytical results presented by some workers suggests that many results taken to indicate the presence of NH3 are experimentally indistinguishable from a result indicating an actual concentration of zero. Indeed, other workers [146] have cited the limit of detection (LOD) of ammonia concentration as 10/igL-1 (i.e., 0.6pM) by the indophenol method and 20/igL 1 (i.e., 1.2pM) by Nessler s method. The ammonia concentrations measured in many photosynthesis experiments hover around these LOD values. 

Values of the second virial coefficient of ethylene for temperatures between 0° and 175°C have been determined to an estimated accuracy of 0.2 cm3/mol or less from low-pressure Burnett PVT measurements. Our values, from —167 to —52 cm3/mol, agree within an average of 0.2 cm3/mol with those recently obtained by Douslin and Harrison from a distinctly different experiment. This close agreement reflects the current state of the art for the determination of second virial coefficient values. The data and error analysis of the Burnett method are discussed. 

This chapter discusses statistics at the level needed for radiation measurements and analysis of their results. People who perform experiments need statistics for analysis of experiments that are statistical in nature, treatment of errors, and fitting a function to the experimental data. The first two uses are presented in this chapter. Data fitting is discussed in Chap. 11. 

For an experiment designed to measure aspect ratio, the error analysis of the previous section shows that for the optimum linear configuration, the... 

The normal distribution was first introduced by de Moivre, who approximated binomial distributions for large n (Figure D.3). His work was extended by Laplace, who used the normal distribution in the error analysis in experiments conducted. Legendre came up with the method of least squares. Gauss, by 1809, justified the normal distribution for experimental errors. The name bell curve was coined by Gallon and Lexis. 

Another sign of development is that a growing number of thermokinetic parameters are accompanied by estimations of their accuracy. In the best cases, these uncertainty estimates are based not on the error analysis of a single experiment but reflect the comparison of several independent experimental or theoretical studies and therefore incorporate systematic errors of the various methods. In the past, such evaluations were performed by human experts, but as the dataset grows, perhaps a new paradigm for this process is required. Data collaboration approaches have been suggested which could place this task in the hands of wide communities (and computer software), rather than small groups of experts. 

In this experiment the overall variance for the analysis of potassium hydrogen phthalate (KHP) in a mixture of KHP and sucrose is partitioned into that due to sampling and that due to the analytical method (an acid-base titration). By having individuals analyze samples with different % w/w KHP, the relationship between sampling error and concentration of analyte can be explored. 

For exposure of reasons of observable discrepancy of results of the analysis simulated experiment with application synthetic reference samples of aerosols [1]. The models have demonstrated absence of significant systematic errors in results XRF. While results AAA and FMA depend on sort of chemical combination of an elements, method of an ashing of a material and mass of silicic acid remaining after an ashing of samples. The investigations performed have shown that silicic acid adsorbs up to 40 % (rel.) ions of metals. The coefficient of a variation V, describing effect of the indicated factors on results of the analysis, varies %) for Mn and Fe from 5 up to 20, for Cu - from 10 up to 40, for Pb - from 10 up to 70, for Co the ambassador of a dry ashing of samples - exceeds 50. At definition Cr by a method AAA the value V reaches 70 %, if element presences an atmosphere in the form of Cr O. At photometric definition Cr (VI) the value V is equal 40%, when the element is present at aerosols in the form of chromates of heavy metals. 

Experience gained in the ZAF analysis of major and minor constituents in multielement standards analyzed against pure element standards has produced detailed error distribution histograms for quantitative EPMA. The error distribution is a normal distribution centered about 0%, with a standard deviation of approximately 2% relative. Errors as high as 10% relative are rarely encountered. There are several important caveats that must be observed to achieve errors that can be expected to lie within this distribution ... 

Human reliability [lata NJUREG/CR-1278 was supplemented by judgment of system analysts and plant personnel. Human error probabilities were developed from NUREG/CR-12 8, human action time windows from system analysis and some recovery limes from analysis of plant specific experience. Data sources were WASH-1400 HEPs,Fullwood and Gilbert assessment ot I S power reactor Bxp., NUREG/ CR -127K. and selected acro ptice li.it.j... 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

The intention of this chapter has been to provide an overview of analytical methods for predicting and reducing human error in CPI tasks. The data collection methods and ergonomics checklists are useful in generating operational data about the characteristics of the task, the skills and experience required, and the interaction between the worker and the task. Task analysis methods organize these data into a coherent description or representation of the objectives and work methods required to carry out the task. This task description is subsequently utilized in human error analysis methods to examine the possible errors that can occur during a task. 

Fitts, P. M., Jones, R. E. (1947). Analysis of Factors Contributing to 460 "Pilot Error" Experiences in Operating Aircraft Controls. Reprinted in H. W. Sinaiko (Ed.) (1961), Selected Papers on Human Factors in the Design and Use of Control Systems. New York Dover. 

FIGURE 11.3 One-way ANOVA (analysis of variance). One-way analysis of variance of basal rates of metabolism in melanophores (as measured by spontaneous dispersion of pigment due to G,.-protein activation) for four experiments. Cells were transiently transfected with cDNA for human calcitonin receptor (8 j-ig/ml) on four separate occasions to induce constitutive receptor activity. The means of the four basal readings for the cells for each experiment (see Table 11.4) are shown in the histogram (with standard errors). The one-way analysis of variance is used to determine whether there is a significant effect of test occasion (any one of the four experiments is different with respect to level of constitutive activity). 

The first group of experiments showed that no significant drift was present the standard deviation (s = 468 counts) and 1he standard counting error (sc — 491 counts) were virtually identical. An analysis of variance for the second and third groups of experiments is summarized in Table 10-6. 

It should be mentioned that the results in Table 10-6 were obtained only after experience had taught that the adjustment drum must be pressed inward for the most precise results. Early trials in which such pressure was not exerted gave reset errors ten times as large. The value of the analysis of variance is thus proved. 

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