Experimental error, random

We also discuss the analysis of the accuracy of experimental data. In the case that we can directly measure some desired quantity, we need to estimate the accuracy of the measurement. If data reduction must be carried out, we must study the propagation of errors in measurements through the data reduction process. The two principal types of experimental errors, random errors and systematic errors, are discussed separately. Random errors are subject to statistical analysis, and we discuss this analysis. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

If the experimental error is random, the method of least squares applies to analysis of the set. Minimize the sum of squares of the deviations by differentiating with respect to m. 

When an analyst performs a single analysis on a sample, the difference between the experimentally determined value and the expected value is influenced by three sources of error random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to determine the effect of random error on the analysis. The position of the distribution relative to the sample s true value, p, is determined both by systematic errors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. 

Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

The sampling of solution for activity measurement is carried out by filtration with 0.22 pm Millex filter (Millipore Co.) which is encapsuled and attached to a syringe for handy operation. The randomly selected filtrates are further passed through Amicon Centriflo membrane filter (CF-25) of 2 nm pore size. The activities measured for the filtrates from the two different pore sizes are observed to be identical within experimental error. Activities are measured by a liquid scintillation counter. For each sample solution, triplicate samplings and activity measurements are undertaken and the average of three values is used for calculation. Absorption spectra of experimental solutions are measured using a Beckman UV 5260 spectrophotometer for the analysis of oxidation states of dissolved Pu ions. 

In order to formulate the statistical problem generally, let us return to the Arrhenius graph (Figure 5) and ask the question of how to estimate the position of the common point of intersection, if it exists (162). That is, in the coordinates x = T and y = log k, a family of 1 straight lines is given with the slopes bj (i = 1,2,..3) and with a common point of intersection (xq, yo). The ith line is determined by mj points (m > 2) with coordinates (xy, yjj) where j = 1,2,..., mj. Instead of the true coordinates yy, only the values yy = yy + ey are available, ey being random variables with a zero average value and a constant variance,. If the hypothesis of a common point of intersection is accepted, ey may be identified with the experimental error. 

The synthetic data have been obtained by adding random noise with standard deviation of about 0.4 )0.g 1 to the theoretical plasma concentrations. As can be seen, the agreement between the estimated and the computed values is fair. Estimates tend to deteriorate rapidly, however, with increasing experimental error. This phenomenon is intrinsic to compartmental models, the solution of which always involves exponential functions. 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

The above equations hold at equilibrium. However, when the measurements of the temperature, pressure and mole fractions are introduced into these expressions the resulting values are not zero even if the EoS were perfect. The reason is the random experimental error associated with each measurement of the state variables, Thus, Equation 14.18 is written as follows... 

The closeness of agreement between independent test results obtained by applying the experimental procedure under stipulated conditions. The smaller the random part of the experimental errors which affect the results, the more precise the procedure. A measure of precision (or imprecision) is the standard deviation. 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

A glass fiber mat in which the fibers appear to be randomly oriented is impregnated with a thermosetting resin and cured. Strips are cut from the sheet in different directions, and their Young s modulus is measured. The Young s moduli are not the same in different directions. If the differences are much greater than the expected experimental errors, what is the most probable cause of the difference in moduli ... 

Figure 16.6 Calibration of the radiocarbon ages of the Cortona and Santa Croce frocks the software used[83] is OxCal v.3.10. Radiocarbon age is represented on the y axis as a random variable normally distributed experimental error of radiocarbon age is taken as the sigma of the Gaussian distribution. Calibration of the radiocarbon agegivesa distribution of probability that can no longer be described by a well defined mathematical form it is displayed in the graph as a dark area on the x axis...

Experimental errors come from two different sources, termed systematic and random errors. However, it is sometimes difficult to distinguish between them, and many experiments have a combination of both types of error. 

The absorption spectra of Aspt, Ace-K, Caf and Na-Benz were recorded from 190 to 300 nm. The calibration set was generated by a three-level full factorial design (4).The absorbance valnes were recorded eveiy 5 nm. The calibration samples were measured in random order, so that experimental errors due to drift were not introduced. 

Nimmo lA, Mabood SF. 1979. The nature of the random experimental error encountered when acetylcholine hydrolase and alcohol dehydrogenase are assayed. Anal Biochem... 

We can tabulate values of A(A//j/(A 2), A(AA//)/(A 2). and A(AAA//)/A 2 and find that the third quantity varies randomly about zcto, which is a behavior that indicates that the second derivative of A/f with respect to U2 is constant within experimental error, or we can fit the data to polynomials of successively higher powers until no significant improvement in standard deviation occurs for the coefficients. 

The experimental desorption isotherms for randomly methylated -cyclodextrin (RAMEB) and RAMEB-enriched minerals are presented in Figure 1. The isotherms were measured for the RAMEB in the forms of powder and crystals however, these were the same within the range of experimental error. (From Jozefaciuk et al., 2001)... 

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