Reduction and the Propagation of Errors

There is a test called the Q test, in which Q is defined as the difference between an outlying data point and its nearest neighbor divided by the difference between the highest and the lowest values in the set  

An outlying data point is discarded if its value of Q of exceeds a certain critical value, which depends on the number of members in the sample. Table 11.2 contains the critical value of Q at the 95% confidence level for samples of N members.  

EXERCISE 11.5 Apply the Q test to the 39.75 °Cdata point appended to 

Many values that are obtained by measurement in a physical chemistry laboratory are used along with other values to calculate some quantity that is not directly measured. Such a calculation is call data reduction. An experimental error in a measured quantity will affect the accuracy of any quantity that is calculated from it. This is called propagation of errors. 

Assume that we have measured two quantities, a and b, and have established a probable value and a 95% confidence interval for each  

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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. 

We also discussed graphical and numerical data reduction procedures. The most important numerical data reduction procedure is the least squares, or regression, method, which finds the best member of a family of functions to represent a set of data. We discussed the propagation of errors through this procedure and presented a version of the procedure in which different data points are given different weights, or importances, in the procedure. 

If the reduction of C—C BDEs by captodative substitution is interpreted with the appropriate caution, it can be stated that a conclusive answer as to the existence of a captodative effect in free radicals cannot be derived from these studies, If, furthermore, a consequent error-propagation analysis had been carried out, the outcome might have been that the error limits do not allow a definitive conclusion. However, the results convey a feeling that— regardless of the pros and cons for the different determination procedures— a possible captodative effect will not be great. 

The inherent difficulty in the measurement of the complex dynamic moduli of viscoelastic materials is emphasized by the results of this paper. The agreement among the shifted modulus data as measured by different systems is limited by several difficulties (1) measurement inaccuracies of the instruments, (2) differences in the data reduction techniques used to apply the time-temperature superposition principle and propagation of shift curve errors and, (3) nonuniformity of the test samples. 

This procedure will not be discussed further here, but the DRG methodology has proven to be easy to implement and to fully automate. Hence, it has become very popular to use for reduction of larger mechanisms. In its simplest form DRG has however proved to have some limitations. These have been addressed in extended versions of the procedure. This includes DRG with error propagation (DRGEP) (Pepiot-Desjardins Pitsch, 2008), DRG aided sensitivity analysis (DRGASA) (Zheng et al., 2007) and DRGEP and sensitivity analysis (Niemeyer et al., 2010). The reader is referred to these works for further discussion. 

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