Weighting of data

These calculations are not, however, all of the same precision, and a simple arithmetic mean will give a disproportionately high weight to the less reliable points. If estimates can be made of the relative precision for each pair of points, then for completely random errors the calculated values of k should be weighted for averaging according to their expected errors. For example, a value having an estimated error of 2 per cent 

In the extreme case that all the time intervals are equal this reduces to 

Similar results hold for reactions of other orders. This is not necessarily a bad result. 

Example, Let us suppose we have three experimentally calculated points together with their estimated errors as listed in the following table. 

It is interesting to observe that the integral method of computing kx generally places most weight on the initial concentration, since that quantity appears in the integrated rate equations (see Table II.2). Such a practice is wise only when the concentration is known with better precision than any of the others. If there is some other point for which the analytical data are better, then it is best to recast the equations in a form which refers the other concentrations to this point (i.e., a shift in coordinate axes). 

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Variance (cont) of prediction, 167 Variance scaling, 100, 174 Vectors basis, 94 Weighting of data, 100 Whole spectrum method, 71 x-block data, 7 x-data, 7 XE, 94 y-block data, 7 y-data, 7... 

It is possible to produce some liquid hydrocarbons from most coals during conversion (pyrolysis and hydrogenation/ catalytic and via solvent refining)/ but the yield and hydrogen consumption required to achieve this yield can vary widely from coal to coal. The weight of data in the literature indicate that the liquid hydrocarbons are derived from the so-called reactive maceralS/ i.e. the vitrinites and exinites present (7 8 1 9). Thusf for coals of the same rank the yield of liquids during conversion would be expected to vary with the vitrinite plus exinite contents. This leads to the general question of effect of rank on the response of a vitrinite and on the yield of liquid products and/ in the context of Australian bituminous coals, where semi-fusinite is usually abundant/ of the role of this maceral in conversion. 

Proper weighting of data to minimize the effect of an error in a single point... 

The characteristic of a relational database model is the organization of data in different tables that have relationships with each other. A table is a two-dimensional consti uction of rows and columns. All the entries in one column have an equivalent meaning (c.g., name, molecular weight, etc. and represent a particular attribute of the objects (records) of the table (file) (Figure 5-9). The sequence of rows and columns in the tabic is irrelevant. Different tables (e.g., different objects with different attributes) in the same database can be related through at least one common attribute. Thus, it is possible to relate objects within tables indirectly by using a key. The range of values of an attribute is called the domain, which is defined by constraints. Schemas define and store the metadata of the database and the tables. 

In this illustration, a Kohonen network has a cubic structure where the neurons are columns arranged in a two-dimensional system, e.g., in a square of nx I neurons. The number of weights of each neuron corresponds to the dimension of the input data. If the input for the network is a set of m-dimensional vectors, the architecture of the network is x 1 x m-dimensional. Figure 9-18 plots the architecture of a Kohonen network. 

The weights of the winning neuron are funher adapted to the input data. The neurons within a certain distance surrounding the winning neuron are also adapted Their weight adaptation is performed such that the closer a neuron is to the winning neuron the more its weights will be adapted. 

Neural networks model the functionality of the brain. They learn from examples, whereby the weights of the neurons are adapted on the basis of training data. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Nimura, Y. Carr, M. R. Reduction of the Relative Error in the Standard Additions Method, Analyst 1990, 115, 1589-1595. The following paper discusses the importance of weighting experimental data when using linear regression Karolczak, M. To Weight or Not to Weight An Analyst s Dilemma, Curr. Separations 1995, 13, 98-104. 

A simpler approach for analyzing neutron activation data is to use one or more external standards. Letting Ao) and (Aq) represent the initial activity for the analyte in an unknown and a single external standard, and letting and represent the weight of analyte in the unknown and the external standard, gives a pair of equations... 

From plots of these data, estimate the Newtonian viscosity of each of the solutions and the approximate rate of shear at which non-Newtonian behavior sets in. Are these two quantities better correlated with the molecular weight of the polymer or the molecular weight of the arms ... 

Figure 8.9 is a plot of osmotic pressure data for a nitrocellulose sample in three different solvents analyzed according to Eq. (8.87). As required by Eq. (8.88), all show a common intercept corresponding to a molecular weight of 1.11 X 10 the various systems show different deviations from ideality, however, as evidenced by the range of slopes in Fig. 8.9. 

Both preparative and analytical GPC were employed to analyze a standard (NBS 706) polystyrene sample. Fractions were collected from the preparative column, the solvent was evaporated away, and the weight of each polymer fraction was obtained. The molecular weights of each fraction were obtained usmg an analytical gel permeation chromatograph calibrated in terms of both and M. The following data were obtained ... 

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