Propagation of error

Often we need to combine several laboratory measurements or do some additional data manipulation to extract specific quantities of interest. Suppose, for example, that you wish to measure the solubility product of silver chloride, which we used in our illustration of quadratic equations in Chapter 1. Consider the following procedure  

The procedure outlined above has many possible sources of both random and systematic error. The measurements of volume and of mass will not be perfectly accurate if the equipment has been correctly calibrated and the laboratory technique is good, these errors are random (equally likely to be positive or negative). The masses of the two nuclei are not perfectly known, but these errors can be assumed to be random as well (the error bars are the results of many careful measurements). The silver chloride and water will both have impurities, which will tend to make systematic errors. Some impurities (e.g., chloride ions in the water) would tend to make the measured solubility product smaller than the true value. Some impurities (e.g., sodium chloride in the silver chloride) would tend to make the measured solubility product larger than the true value. Finally, even without impurities, there is one (probably small) systematic error 

In good laboratory procedure, the systematic errors are significantly smaller than the random errors. Assuming this is the case, the random errors are propagated as follows  

This formula is easy to verify with the coin toss distribution. As discussed above, with 10,000 coin tosses you have 5000 98 heads. You can also verify (by calculating a) that with 2500 coin tosses you have 1250 49 heads, but for 12,500 tosses you get 6250 109.56733 heads, not 6250 147 heads as you would get by just adding the error bars for 10,000 tosses and 2500 tosses. We would report 6250 110 heads, using the rounding off convention discussed above (last digit uncertain between 3 and 30 units). 

In propagating errors it is generally advisable to keep one or two extra digits in intermediate results, and round off only when you get to the final result. Applying this rule to the solubility product measurement, the mass of dissolved silver chloride is 0.888 0.0142 mg, and the formula weight of silver chloride is 143.3279 0.00042 g- mol 1. 

An experiment has been performed a variety of direct measurements have been made—weights, volumes, temperatures, measured electromotive forces, spectral absorbances, etc.—and uncertainties in all of them have been estimated, either from statistical data obtained by repeated measurements or from judgment and experience. From the values obtained by direct measurement and with the aid of a phenomenological theory, a final numerical result is calculated. Let the desired numerical result be designated by Fand the directly measured quantities be designated yy x,y, z,. The latter quantities are assumed to be mutually independent. Let their uncertainties, usually in the form of 95 percent confidence limits, be designated A(.t), A( ), A(. .The value of Fis determined by sub- 

The issue to be discussed here is how one can estimate the uncertainty of the final result F, usually in the form of a 95 percent confidence limit. 

Infinitesimal changes dx, dy, etc., in the experimentally determined values will produce in Fthe infinitesimal change dF, where 

If the changes are finite rather than infinitesimal, but are small enough that the values of the partial derivatives are not appreciably affected by the changes, we have approximately 

This is equivalent to a Taylor expansion in which only the first-power terms have been retained. Now suppose that Ax represents the experimental error ( in the quantity x  

For the estimation of the errors associated with counting and the calculation of activities we refer to the literature on the measurement of radiation like Tsoulfanidis (1995) and Ivanovich and Hannon (1992). Here we will only mention the main principles  

The standard deviation associated with the measurement of N disintegrations is  

If a physical magnitude Y is to be obtained by summation or difference of independent observations with errors r, the error R in F is found by  

An observed count rate consists of a contribution from the background and a contribution from the somce. The error in the net count rate (s) is given by 

The standard deviation, therefore, is the square root of the sum of the squares of the component standard deviations. 

21) can be used only if the variables K, Y2.Yn are independent or if their uncertainties are small, that is the covariances can be disregarded. One of these two assumptions can almost always be made in chemical thermodynamics, and Eq. (C.21) can thus almost universally be used in this review. Eqs. (C.22) through (C.26) present explicit formulas for a number of frequently encountered algebraic expressions, where c, ci, C2 are constants. 

A few simple calculations illustrate how these formulas are used. The values have not been rounded. 

it can be seen that the uncertainty in log,p K° cannot be the same as in In K°. The constant conversion factor of In(lO) = 2.303 is also to be applied to the uncertainty. 

When the digit following the last digit to be retained is less than 5, the last digit retained is kept unchanged. 

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Donato, H. Metz, G. A Direct Method for the Propagation of Error Using a Personal Gomputer Spreadsheet Program, ... 

If the errors in the variables are independent, then = 0, and the propagation of error equation can be written... 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

We apply the propagation of errors treatment to Eq. (2-93), where the quantities in parentheses are treated as known constants. The result is... 

For an example of curve fitting involving classical propagation of errors in a potentiometric titration setting, see Ref. 142. 

A variation of the lUPAC method called the propagation of errors (PE) method has been discussed by Long and Winefordner. In the PE method, the LOD is defined as... 

A better alternative would be to use the propagation of errors definition, which takes into consideration values of both and si when calculating the MDL. This would involve generating at least five calibration curves in order to obtain an accurate measurement of si and Sm ... 

Finally, Chapter 16 provides information about the handling of U-series data, with a particular focus on the appropriate propagation of errors. Such error propagation can be complex, especially in the multi-dimensional space required for U- " U- °Th- Th chronology. All too often, short cuts are taken during data analysis which are not statistically justified and this chapter sets out some more appropriate ways of handling U-series data. 

In derivatisation reactions with a KIE correction factors are first calculated this calculation introduces another step where errors propagate. The propagation of errors under these circumstances is calculated using Equation (14.5), where subscript s stands for the standard used in correction factor determination and sd stands for the derivatised standard. The magnitude of the errors associated with the correction factors themselves can be calculated using Equation (14.4), along with the precisions for each determination (Docherty et al. 2001). 

Applying equation (4.3.4) for the linear propagation of errors gives the covariance matrix Sy of the vector... 

Opeea, T.L, Olah, M., Ostopovici, L., Rad, R., and Meacec, M. On the propagation of errors in the QSAR literature. In EuroQSAR 2002 - Designing Drugs and Crop Protectants Processes, Problems and Sdutions, Foed, M., Livingstone, D., Deaeden, J., and... 

If we combine some standard nncertainty contributions from different sonrces then we calcnlate that according to the law of propagation of errors (sqnare root of the snm of sqnares) and call it combined (standard) nncertainty. 

The general relationship between the combined standard uncertainty Uc(y) of a value y and the uncertainty arising from the independent parameters x. X2.. ..Xn on which it depends is according to the law of propagation of errors... 

The scaled total electron density pobs = p obs/k, where p obs is the density on the experimental scale. Using the expression for propagation of errors given in Eq. 

Propagation of error Mathematical technique for computing the total error of a model prediction by calculating the error for each term in the model, and propagating the errors through the model into the total error of prediction. 

The variance for the weight fraction w(i) can be obtained from equation 1H and the variance for M(i) can be calculated directly from equation 18. Note that the propagation of error analysis can be readily extended to other averages and and it can also be used to account for the errors associated with the calibration of columns and detectors. 

Using the rules of the propagation of errors [33,34] a measure of the robustness of the partition coefficient (C, ) and the robustness of the selectivity (C ) can be obtained. Below, a derivation of robustness of the partition coefficient P, of a compound i and the selectivity Uij for two compounds i and j with respect to variation in extraction liquid composition is given. The general form of a (Special Cubic) mixture model for three-component mixtures is given by ... 

Accuracy and precision depend on the propagation of error starting from the error in weighing, volumetrically preparing the sample, and delivering the titrant to the sample. 

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