Propagation of random errors

In experimental work, the quantity to be determined is often calculated from a combination of observable quantities. We have already seen, for example, that even a relatively simple operation such as a titrimetric analysis involves several stages, each of which will be subject to errors (see Chapter 1). The final calculation may involve taking the sum, difference, product or quotient of two or more quantities or the raising of any quantity to a power. 

It is most important to note that the procedures used for combining random and systematic errors are completely different. This is because random errors to some extent cancel each other out, whereas every systematic error occurs in a definite and known sense. Suppose, for example, that the final result of an experiment, x, is given by x = a + b. If a and b each have a systematic error of -i-l, it is clear that the systematic error in x is +2. If, however, a and b each have a random error of 1, the random error in x is not 2 this is because there will be occasions when the random error in a is positive while that in b is negative (or vice versa). 

This section deals only with the propagation of random errors (systematic errors are considered in Section 2.12). If the precision of each observation is known then simple mathematical rules can be used to estimate the precision of the final result. These rules can be siunmarized as follows. 

In this case the final value, y, is calculated from a linear combination of measured quantities a, b, c, etc., by  

In a titration the initial reading on the burette is 3.51 ml and the final reading is 15.67 ml, both with a standard deviation of 0.02 ml. What is the volume of titrant used and what is its standard deviation  

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In certain cases, the propagation of random errors can be carried out very simply ... 

Propagation of Random Errors. Let x, y, z,. .. be independent, direcdy measured quantities let Fbe a result calculated from them with an equation represented symbolically by... 

After the reconcilation is effected, adjusted values are substituted into equations of the horizontal band 1, and the unmeasured values (vertical band 1) are simply computed. In a similar way, also a complex processing of the measured data (propagation of random errors, detection of gross errors, etc.) can be realized see again Chapter 9, or for example Madron (1992). Measured data inconsistency is analyzed using the equations from bands 3 and 4a only. The remaining equations contain no information in this respect. 

This test code specifies procedures for evaiuation of uncertainties in individuai test measurements, arising from both random errors and systematic errors, and for the propagation of random and systematic uncertainties... 

By careful proceeding of measurements random variations can be minimized, but fundamentally not eliminated. The appearance of random errors follow a natural law (often called the Gauss law ). Therefore, random variations may be characterized by mathematical statistics, namely, by the laws of probability and error propagation. 

This part of the chapter is concerned with the evaluation of nncertainties in data and in calculated results. The concepts of random errors/precision and systematic errors/accuracy are discussed. Statistical theory for assessing random errors in finite data sets is summarized. Perhaps the most important topic is the propagation of errors, which shows how the error in an overall calculated result can be obtained from known or estimated errors in the input data. Examples are given throughout the text, the headings of key sections are marked by an asterisk, and a convenient summary is given at the end. 

Recall that the precision quantifies random errors. Let s consider the titration curve especially about the equivalence point (Fig. 10.1). It appears that the lower the slope is at the equivalence point, the higher the propagation of the error due to the pH measurement is. This is sufficient to justify the introduction of the parameter Ti to quantify the phenomenon, q is named the sharpness index. It is defined as being the magnitude of the slope of the titration curve ... 

The formal linear propagation of random measurement errors into /(z) can be written... 

A final caution concerns the error introduced into the calculated interproton distances. As this depends on the errors of the measured quantities, it is propagated through the calculations according to Eq. for independent and random errors, namely,... 

It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are defined by random errors. The total deviation of an analytical measurement, frequently called the total analytical error , is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule. 

For variables vAiose errors are not Independent, a general form of Equation 11 Incorporates covarleince as well as variance. Also, another form of the equation has been derived for non-random error (20). As will be seen below, certain ccxnputatlons, well known In SBC, cure now being found to be Intrinsically Imprecise because of error propagation. 

Since the uncertainties in and > 238 independent and random, then the uncertainty in the age of the zircon can be calculated using error propagation for independent random errors as... 

By far, most propagation of uncertainty computations that you will encounter deal with random error, not systematic error. Our goal is always to eliminate systematic error. 

This discussion deals with random errors and their propagation in reported HO concentrations. Equal attention should be given, of course, to systematic errors of calibration or instrument drift. 

Owing to its gradient-descent nature, back-propagation is very sensitive to initial conditions. The choice of initial weights will influence whether the net reaches a global (or only a local) minimum of the error and, if so, how quickly it converges. In practice, the weights are usually initialized to small zero-mean random values between -0.5 and 0.5 (or between -1 and 1 or some other suitable interval). 

An error analysis dealing with the uncertainty in the final result due to random errors in the measurements will normally be part of the Results section. The type of error analysis undertaken will depend a great deal on the nature of the experiment see Chapters IIB and XXI for more details. The analysis given in the sample report is typical of a straightforward propagation-of-errors treatment. If a long and complex propagation-of-errors... 

The estimation of the error of a computed result R from the errors of the component terms or factors A, B, and C depends on whether the errors are determinate or random. The propagation of errors in computations is summarized in Table 26-2. The absolute determinate error e or the variance V = s for a random error is transmitted in addition or subtraction. (Note that the variance is additive for both a sum and a difference.) On the other hand, the relative determinate error ejx or square of the relative standard deviation (sJxY is additive in multiplication. The general case R = f A,. ) is valid only if A, B,C,... are independently variable it is... 

Hyphenated multidimensional analytical instrumentation requires careful calibration and maintenance to obtain high quality, meaningful data (J9). Because of the propagation of systematic and random errors as different analytical instrumentation are interfaced, frequent calibration using well-characterized polymer standards is required even for absolute M -sensitive detectors. Eurthermore, the relatively low signal-to-noise ratio at the ends of the MWD can lead to significant uncertainties in these regions of the distribution unfortunately, these areas of the distribution can profoundly affect polymer properties. 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

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