Solution to measurement uncertainty challenge

Analytical and Bioanalytical Chemistry, Sep 2017

Juris Meija

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Solution to measurement uncertainty challenge

The winner of the measurement uncertainty challenge (pub- lished in Solution to measurement uncertainty challenge Juris Meija 0 Option A 0 0 Measurement Science and Standards, National Research Council Canada , 1200 Montreal Road M-12, Ottawa, ON K1A 0R6 , Canada The analysis of sterling silver has produced information regarding the silver content in this alloy [1]. Although we are interested only in the mass fraction of silver in the alloy, let us first collect all that is known about this material: 1. The alloy is produced from pure silver and pure copper. Consequently, no other elements are expected in the final material. 2. The mass of each pin is 1 g. 3. Determination of silver from five pins gave the following results for the mass of silver in each pin: m(Ag) = 0.844 g, 0.888 g, 0.825 g, 0.907 g, 0.882 g, with standard deviation s(m(Ag)) = 0.034 g. 4. Determination of copper from five pins gave the following results for the mass of copper in each pin: m(Cu) = 0.060 g, 0.096 g, 0.067 g, 0.075 g, 0.070 g, with standard deviation s(m(Cu)) = 0.014 g. - This model provides rather a straightforward estimate of the mass fraction of silver: w(Ag) = m(Ag)/(1 g). The mean of the five observations is wA(Ag) = 0.869 g/g. The 95% confidence interval of wA(Ag) can be obtained, assuming a normal distribution of the measurement errors. Since the actual (true) magnitude of the measurement errors is unknown, we use the quantiles of the t distribution with four degrees of freedom to obtain the expanded uncertainty of wA(Ag): wCðAgÞ ¼ mðAgmÞðþAgmÞðCuÞ : ð5Þ ð6Þ U ðwAðAgÞÞ ¼ tp¼1−0:205;v¼4 ¼ 0:042 g=g: sðmpðAffi5ffi gÞÞ ¼ 2:78 0p:03ffi5ffi4 siðmðAgÞÞ ¼ sðmðAgÞÞ Alternatively, the doubt about the standard deviation s(m(Ag)) can be incorporated with use of the parametric bootstrap resampling method, where we draw samples from a normal distribution with a known mean (0.869 g/g) and unknown standard deviation. Here, the standard deviation itself is modeled as a random variable that follows a scaled and shifted chi-squared distribution to be consistent with the observed standard deviation: sffiffiffiffiffiffiffiffiffi N −1 ð2Þ χ2N−1 : We proceed with random sampling (of N = 5 samples) from the normal distribution with the mean 0.869 g/g and standard uncertainty si as given in Eq. 2. The means of such samples provide 95% confidence interval U(wA(Ag)) = 0.042 g/g, in agreement with the result shown in Eq. 1. Option B Option C Calculations for this model follow the same route as in option A, and provide wB(Ag) = 0.926 g/g with expanded uncertainty U(wB(Ag)) = 0.017 g/g with use of either Eq. 1 or Eq. 2. To some, the most intuitive choice for data reduction of this dataset is to take into account both silver and copper determinations while disregarding the fact that the pins each weigh 1 g. The corresponding measurement model is ð1Þ Because the same variable appears in the equation, one can rewrite Eq. 3 to avoid the dependency problem between the numerator and the denominator: wCðAgÞ ¼ 1=ð1 þ mðCuÞ=mðAgÞÞ: From Eq. 4, we obtain wC(Ag) = 0.922 g/g. Modeling the average results for both m(Ag) and m(Cu) as random variables with a t distribution, m(Ag) ~ t(μ = 0.869, s = 0.015, v = 4) and m(Cu) ~ t(μ = 0.074, s = 0.006, v = 4), we obtain expanded uncertainty U(wC(Ag)) = 0.012 g/g. The parametric bootstrap method involving Eq. 2 to account for the uncertainty in the standard deviation of both copper and silver measurements provides an expanded uncertainty U(wC(Ag)) = 0.017 g/g. Peter mistakenly thought that both silver and copper measurements were made from the same five pins. He calculated the correlation between these data (0.578) and propagated the uncertainty by taking this into account (The NIST Uncertainty Machine at http://uncertainty.nist.gov is useful to do such calculations). Option D Option D combines all the available information while assuming the homogeneity of the pins. This can be achieved, for example, by our taking the average of the results wA and wB. For this purpose, we can choose the weighted average: wDðAgÞ ¼ u−2ðwAÞ⋅wAðAgÞ þ u−2ðwBÞ⋅wBðAgÞ u−2ðwAÞ þ u−2ðwBÞ ; 1 u2ðwDðAgÞÞ ¼ u−2ðwAÞ þ u−2ðwBÞ : Using the values wA(Ag) = 0.869(15) g/g and wB(Ag) = 0.926(6) g/g, we obtain wD(Ag) = 0.918(6) g/g. The expanded uncertainty of wD(Ag) can be obtained by multiplication of the standard uncertainty and the corresponding t value. The Welch–Satterthwaite equation provides the approximate degree of freedom for the weighted average, v = 5.2, which corresponds to a 95% critical t value of 2.6 and expanded uncertainty U(wD(Ag)) = 2.6 × 0.006 = 0.016 g/g. The weighted average is not without its shortcomings. Most notably, as explained earlier, the estimates of variance can be inaccurate when they are derived from a small number of replicates. Option E Option D assumes that all pins have identical mass fraction of silver, which might not be the case. A method that does not make such an assumption is nonparametric bootstrap resampling of the data. For this purpose, we can evaluate the uncertainty of the silver mass fraction from bootstrap resampling (with replacement) from the available set of ten results: five values of m(Ag)/(1 g) and five values of (1 – m(Cu))/(1 g), which treats both silver and copper data on an equal footing. The average is calculated for each bootstrap sample and, after a sufficient number of such samples, the following summary statistics is obtained: wE(Ag) = 0.898 g/g with expanded uncertainty U(wE(Ag)) = 0.023 g/g. Summary In summary, different choices on how to interpret the measurement results will lead to different uncertainty statements, [ 2 ] as shown in Table 2. The approaches outlined here are by no means the only choices that can be made, and many Expanded uncertainty, U(w(Ag)) 0.042 g/g 0.017 g/g 0.017 g/g 0.016 g/g 0.023 g/g readers might choose other ways to interpret this seemingly simple dataset. Indeed, as is noted in the Guide to the Expression of Uncertainty in Measurement, “the evaluation of measurement uncertainty is neither a routine task nor a purely mathematical one” [ 3 ]. Hence, measurement uncertainty is somewhat of a personal statement; it is your uncertainty, not the uncertainty. 1. Meija J . Measurement uncertainty challenge . Anal Bioanal Chem . 2017 ; 409 : 2497 . 2. Possolo A , Pintar A L . Plurality of Type A evaluations of uncertainty . Metrologia 2017 ; 54 : 617 . 3. Joint Committee for Guides in Metrology. Evaluation of measurement data - Guide to the expression of uncertainty in measurement , JCGM 100 : 2008 . Joint Committee for Guides in Metrology; 2008


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Juris Meija. Solution to measurement uncertainty challenge, Analytical and Bioanalytical Chemistry, 2017, 5799-5801, DOI: 10.1007/s00216-017-0540-2