Input data of the examples of Annex H of the GUM (JCGM 100:2008), of clause 9 of its supplements (JCGM 101:2008, JCGM 102:2011), of NIST TN 1900 and of the EURACHEM/CITAC Guide QUAM, in the formats used by propagate, propagateMV and makeDat. The examples are evaluated in the demos demo("JCGM100") (H.1 to H.6), demo("JCGM101") (S1.9.3, S1.9.5), demo("JCGM102") (S2.9.4, S2.9.5), demo("NIST1900") (TN.E2, TN.E11) and demo("QUAM") (QUAM.A4, QUAM.A5); H.1 is also used in demo("JCGM106") for a conformity assessment according to JCGM 106:2012 (see conformity).
data(H.1); data(H.2); data(H.3); data(H.4); data(H.5); data(H.6)
data(S1.9.3); data(S1.9.5)
data(S2.9.4); data(S2.9.5)
data(TN.E2); data(TN.E11); data(QUAM.A4); data(QUAM.A5)H.1GUM H.1, end-gauge calibration, Table H.1: a matrix with the rows estimate, u and nu and the columns Ls (length of the standard, nm), d (difference in length, nm), da (difference of the thermal expansion coefficients, 1/degC), theta (deviation from 20 degC, degC), as (thermal expansion coefficient of the standard, 1/degC) and dt (temperature difference, degC); for propagate(type = "stat"). Model (GUM H.3): \(l = l_S + d - l_S(\delta\alpha\,\theta + \alpha_S\,\delta\theta)\).
H.2GUM H.2, simultaneous resistance and reactance measurement, Table H.2: a data frame with five sets of simultaneous indications of the amplitude V (V) of a sinusoidally-alternating potential difference, the amplitude I (A) of the alternating current and the phase-shift angle phi (rad). The measurands are related to the inputs by Ohm's law (GUM H.7): \(R = (V/I)\cos\phi\), \(X = (V/I)\sin\phi\), \(Z = V/I\); since \(Z^2 = R^2 + X^2\), there are only two independent output quantities. The indications are correlated and evaluated with type = "raw", cov = "full".
H.3GUM H.3, calibration of a thermometer, Table H.6: a data frame with \(n = 11\) thermometer readings tk (degC, negligible uncertainty) and the measured corrections bk = \(t_{R,k} - t_k\) (degC) against reference temperatures in the range 21 degC to 27 degC. A linear calibration curve \(b(t) = y_1 + y_2(t - t_0)\), \(t_0\) = 20 degC (GUM H.12), is fitted by least squares; intercept and slope, with their variances and covariance, predict the correction and its uncertainty at any temperature \(t\), e.g. with predictNLS.
H.4GUM H.4, measurement of activity, Table H.7: a data frame with six cycles of counting of a standard, a blank and a sample: the times ts, tb, tx (min) from the reference time to the midpoints of the dead-time-corrected counting intervals \(T_0\) = 60 min, and the numbers of counts Cs, Cb, Cx. The decay- and background-corrected count rates are \(R = (C - C_B)\,e^{\lambda t}/T_0\) with \(\lambda\) = 1.25894e-4 1/min, and the activity concentration of the sample is \(A_x = A_S (m_S/m_x)(\bar{R}_x/\bar{R}_S)\) (GUM H.19, H.21) with \(A_S\) = 0.1368 Bq/g, \(m_S\) = 5.0192 g, \(m_x\) = 5.0571 g (GUM H.4.3).
H.5GUM H.5, analysis of variance, Table H.9: a data frame with the columns day (day number), Vj (daily mean, V) of \(K = 5\) observations of a Zener voltage standard and s (their standard deviation, microvolt).
H.6GUM H.6, Rockwell C hardness, Table H.10: an uncertainty budget for makeDat (columns name, value, u, a, dist) with the mean indentation depth d, the machine difference Dc, the block variation Db (triangular) and the national standard (column DS, \(\Delta_S\) in the model), in Rockwell units (0.002 mm). Model (GUM H.33a): \(h = 100 - d - \Delta_c - \Delta_b - \Delta_S\).
S1.9.3JCGM 101:2008, 9.3, mass calibration, Table 5: an uncertainty budget for makeDat with the conventional mass mR and the mass difference dmR of the reference weight (mg, Gaussian), and the densities of air ra, of the weight rW and of the reference weight rR (kg/m^3, rectangular). Model (JCGM 101 Eq. 24): \(\delta m = (m_{R,c} + \delta m_{R,c})[1 + (\rho_a - \rho_{a0})(1/\rho_W - 1/\rho_R)] - m_{nom}\), with \(\rho_{a0}\) = 1.2 kg/m^3 and \(m_{nom}\) = 100 g.
S1.9.5JCGM 101:2008, 9.5, gauge block calibration, Table 10: an uncertainty budget for makeDat with the columns name, value, u, df, a, d and dist (t, rectangular, Gaussian, arcsine and curvilinear trapezoidal distributions; lengths in nm). Model (JCGM 101 Eq. 37): \(\delta L = L_s + D + d_1 + d_2 - L_s[\delta\alpha(\theta_0 + \Delta) + \alpha_s\delta\theta] - L_{nom}\), \(L_{nom}\) = 50 mm.
S2.9.4JCGM 102:2011, 9.4, simultaneous resistance and reactance measurement, Table 8: the data of GUM H.2 (H.2) with a sixth set of indications, a data frame with V (V), I (A) and phi (rad).
S2.9.5JCGM 102:2011, 9.5, Celsius temperature with a resistance thermometer: a list with inputs (Table 12: R0 in Ohm, A in 1/degC, B in 1/degC^2, RS in Ohm, with standard uncertainties), cor (Table 13: correlation matrix of R0, A, B) and ratios (Table 14: ten measured resistance ratios \(r_j\) with their standard uncertainty). Implicit model (JCGM 102 Eq. 31): \(R_0(1 + A\theta + B\theta^2) - r R_S = 0\).
TN.E2NIST TN 1900, E2, surface temperature (Exhibit 2): a data frame with the daily maximum temperatures t (degC) read on 22 days of May 2012. Mean 25.59 degC, \(u = s/\sqrt{22}\) = 0.872 degC.
TN.E11NIST TN 1900, E11, microwave step attenuator (EA-4/02, S7): a list with LS, four observations of the attenuation difference (dB; mean 30.0402 dB, \(u = s/\sqrt{4}\) = 0.0091 dB), and budget, the input quantities of Exhibit 16 with estimates, standard uncertainties u (dB) and distributions ("t3": Student t with 3 degrees of freedom, rescaled to standard deviation u; arcsine and rectangular inputs with standard deviation u). Model: \(L_X = L_S + \delta L_S + \delta L_D + \delta L_M + \delta L_K + \delta L_{ib} - \delta L_{ia} + \delta L_{0b} - \delta L_{0a}\).
QUAM.A4EURACHEM QUAM, A4, organophosphorus pesticides in bread, Table A4.2: a data frame with 15 duplicate analyses D1, D2 (mg/kg) of residue, taken in different runs. The standard deviation of the relative differences, 0.382, divided by \(\sqrt{2}\), gives the relative standard uncertainty 0.27 of a single result (intermediate precision). In Table A4.2, the means of rows 13 and 14 are misprinted (0.75 and 0.10 instead of 0.075 and 0.01); the dataset contains the duplicate values only.
QUAM.A5EURACHEM QUAM, A5, cadmium release from ceramic ware, Table A5.2: a data frame with the AAS calibration: three replicate absorbances (column absorbance) at each of five cadmium concentrations conc (mg/L). The least-squares line has slope 0.2410 and intercept 0.0087 with residual standard deviation 0.005486; for the leach solution, measured twice, \(c_0\) = 0.26 mg/L with \(u(c_0)\) = 0.018 mg/L (QUAM, A5 and E.4).
GUM:
Evaluation of measurement data - Guide to the expression of uncertainty in measurement.
JCGM 100:2008. Joint Committee for Guides in Metrology (2008).
tools:::Rd_expr_doi("10.59161/JCGM100-2008E")
GUM Supplements and conformity assessment:
Evaluation of measurement data - Supplement 1 to the "Guide to the expression of uncertainty in measurement" - Propagation of distributions using a Monte Carlo method.
JCGM 101:2008. Joint Committee for Guides in Metrology (2008).
tools:::Rd_expr_doi("10.59161/JCGM101-2008")
Evaluation of measurement data - Supplement 2 to the "Guide to the expression of uncertainty in measurement" - Extension to any number of output quantities.
JCGM 102:2011. Joint Committee for Guides in Metrology (2011).
tools:::Rd_expr_doi("10.59161/JCGM102-2011")
Evaluation of measurement data - The role of measurement uncertainty in conformity assessment.
JCGM 106:2012. Joint Committee for Guides in Metrology (2012).
tools:::Rd_expr_doi("10.59161/JCGM106-2012")
Collections of worked examples:
Compendium of examples: Good practice in evaluating measurement uncertainty (EMUE), version 2.
van der Veen AMH, Cox MG (Eds.).
Zenodo (2021).
tools:::Rd_expr_doi("10.5281/zenodo.5142180")
Quantifying Uncertainty in Analytical Measurement (QUAM:2012.P1), 3rd edition.
Ellison SLR, Williams A (Eds.).
EURACHEM/CITAC Guide CG 4 (2012). ISBN 978-0-948926-30-3.
https://www.eurachem.org/index.php/publications/guides/quam
Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
Possolo A.
NIST Technical Note 1900, National Institute of Standards and Technology (2015).
tools:::Rd_expr_doi("10.6028/NIST.TN.1900")
propagate, propagateMV, makeDat; demo(package = "propagate")
## GUM H.1: l - 50 mm = 838 nm, u = 32 nm
data(H.1)
propagate(expression(Ls + d - Ls * (da * theta + as * dt) - 50000000), H.1,
type = "stat", alpha = 0.01, nsim = 1E4, check = FALSE)
## NIST TN 1900, E2: mean 25.59 degC, u = 0.872 degC, 95 % coverage interval (23.8, 27.4) degC
data(TN.E2)
propagate(expression(t), as.matrix(TN.E2), type = "raw", nsim = 1E4, check = FALSE)
## EURACHEM QUAM, A5: calibration line and u(c0) = 0.018 mg/L for c0 = 0.26 mg/L (two readings)
data(QUAM.A5)
FIT <- lm(absorbance ~ conc, data = QUAM.A5); S <- summary(FIT)$sigma
Sxx <- sum((QUAM.A5$conc - mean(QUAM.A5$conc))^2)
S/coef(FIT)[2] * sqrt(1/2 + 1/15 + (0.26 - mean(QUAM.A5$conc))^2/Sxx)
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