Propagation of uncertainties

The previous section discussed combining the results of different measurements to obtain a better overall result. We noted that data Set B in Table 5.2 must have undisclosed sources of uncertainty. Let us suppose that it becomes apparent that the preparation of the sources had introduced an extra uncertainty of 6.5 % in the case of the first source and 5.3 % for the second. How can we include the information The calculation of the uncertainty for each data item, using the example of Set B, is as follows  

This would provide us with a weighted mean of 10.68 with a pooled uncertainty of 4.94%, consistent with the actual spread of the data suggested by an external uncertainty of 4.68 %. This is an example of propagation of uncertainty. Because the source preparation factor is multiplicative, Equation (5.13) from Section 5.1.1 can be used to combine the uncertainties. The uncertainties are said to have been combined in quadrature. (We will meet this again later when discussing the factors that combine to create the width of gamma-ray peaks.) 

In our example here, if the source preparation uncertainty were a fixed amount for the method it would be an item common to both sources. It should not, therefore, be included when the uncertainties on the individual results are calculated. It should be taken into account by adding in quadrature to the weighted mean result. If, in our example, the sample preparation uncertainty were 6.5 % for both samples, then the overall uncertainty of the weighted mean for Set B would be y (4.41 -1-6.5 ) = 7.85 %. The weighted mean value would be unchanged. 

In a radioactivity measurement, we may have several sources of uncertainty, all of which must be taken into account in our final uncertainty. For example, we might have  

Equation (5.13) can only be used in this way when the various factors are multiplied together. If the factors 

We can usually estimate or measure the random error associated with a measurement, such as the length of an object or the temperature of a solution. The uncertainty might be based on how well we can read an instrument or on our experience with a particular method. If possible, uncertainty is expressed as the standard devia-tion or as a confidence interval based on a series of replicate measurements. The 

In most experiments, it is necessary to perform arithmetic operations on several numbers, each of which has an associated random error. The most likely uncertainty in the result is not simply the sum of the individual errors, because some of them are likely to be positive and some negative. We expect some cancellation of errors. 

Suppose you wish to perform the following arithmetic, in which experimental uncertainties, designated 2, and 3, are given in parentheses. 

The arithmetic answer is 3.06 but what is the uncertainty associated with this result  

For addition and subtraction, the uncertainty in the answer is obtained from the absolute uncertainties of the individual terms as follows  

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So what is the total uncertainty when using this pipet to deliver two successive volumes of solution from the previous discussion we know that the total uncertainty is greater than 0.000 mL and less than 0.012 mL. To estimate the cumulative effect of multiple uncertainties, we use a mathematical technique known as the propagation of uncertainty. Our treatment of the propagation of uncertainty is based on a few simple rules that we will not derive. A more thorough treatment can be found elsewhere. ... 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. 

Many other mathematical operations are commonly used in analytical chemistry, including powers, roots, and logarithms. Equations for the propagation of uncertainty for some of these functions are shown in Table 4.9. 

Given the complexity of determining a result s uncertainty when several measurements are involved, it is worth examining some of the reasons why such calculations are useful. A propagation of uncertainty allows us to estimate an ex-... 

A propagation of uncertainty also helps in deciding how to improve the uncertainty in an analysis. In Example 4.7, for instance, we calculated the concentration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty of 2 ppm and a relative uncertainty of 1.6%. How might we improve the analysis so that the absolute uncertainty is only 1 ppm (a relative uncertainty of 0.8%) Looking back on the calculation, we find that the relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank)... 

A standard solution of Mn + was prepared by dissolving 0.250 g of Mn in 10 ml of concentrated HNO3 (measured with a graduated cylinder). The resulting solution was quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution was pipeted into a 500-mL volumetric flask and diluted to volume, (a) Express the concentration of Mn in parts per million, and estimate uncertainty by a propagation of uncertainty calculation, (b) Would the uncertainty in the solution s concentration be improved... 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

I. 000 X 10- 1.000 X 10-k 1.000 X 10-k and 1.000 X 10- M from a 0.1000 M stock solution. Calculate the uncertainty for each solution using a propagation of uncertainty, and compare to the uncertainty if each solution was prepared by a single dilution of the stock solution. Tolerances for different types of volumetric glassware and digital pipets are found in Tables 4.2 and 4.4. Assume that the uncertainty in the molarity of the stock solution is 0.0002. 

We can derive an expression between precision and transmittance by applying the propagation of uncertainty as described in Chapter 4. To do so we write Beer s law as... 

The effect of an uncertainty in potential on the accuracy of a potentiometric method of analysis is evaluated using a propagation of uncertainty. For a membrane ion-selective electrode the general expression for potential is given as... 

Cabaniss, S., Propagation of Uncertainty in Aqueous Equilibrium Calculations Non-Gaussian Output Distributions, Anal. Chem. 69, 1997, 3658-3664. 

Bauer et al. [1991a] derived the following propagation of uncertainty for x... 

To put equation 44-6 into a usable form under the conditions we wish to consider, we could start from any of several points of view the statistical approach of Hald (see [10], pp. 115-118), for example, which starts from fundamental probabilistic considerations and also derives confidence intervals (albeit for various special cases only) the mathematical approach (e.g., [11], pp. 550-554) or the Propagation of Uncertainties approach of Ingle and Crouch ([12], p. 548). In as much as any of these starting points will arrive at the same result when done properly, the choice of how to attack an equation such as equation 44-6 is a matter of familiarity, simplicity and to some extent, taste. 

Therefore, continuing as we originally did, we note that we, being chemists and spectroscopists, and writing for spectroscopists, will use the Propagation of Uncertainties approach of Ingle and Crouch ... 

Hoffman FO, Hammonds IS. 1994. Propagation of uncertainty in risk assessments the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 14 707-712. 

The major components of uncertainty are combined according to the rules of propagation of uncertainty, often with the assumption of independence of effects, to give the combined uncertainty. If the measurement uncertainty is to be quoted as a confidence interval, for example, a 95% confidence interval, an appropriate coverage factor is chosen by which to multiply the combined uncertainty and thus yield the expanded uncertainty. The coverage factor should be justified, and any assumptions about degrees of freedom stated. 

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. 

Table 3-1 summarizes rules for propagation of uncertainty. You need not memorize the rules for exponents, logs, and antilogs, but you should be able to use them. 

Chapter 3 gave rules for propagation of uncertainty in calculations. For example, if we were dividing a mass by a volume to find density, the uncertainty in density is derived from the uncertainties in mass and volume. The most common estimates of uncertainty are the standard deviation and the confidence interval. 

In the preceding example, an unknown with a corrected absorbance of y = 0.302 had a protein content of = 18.24 pg. What is the uncertainty in the number 18.24 A full treatment of the propagation of uncertainty gives the following results 1-9... 

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