GUM Uncertainty Budgets

1. Theory

• Definition: An uncertainty budget lists every input to a measurement, assigns each a standard uncertainty, and combines them through the measurement model into a combined and an expanded uncertainty (GUM; NIST TN 1297).
• Type A and Type B: Type A components come from statistics of repeated observations, Type B from other information such as specifications, calibration certificates and assumed distributions (NIST TN 1297).
• Propagation: Sensitivity coefficients and correlations carry input uncertainties into the result (GUM; NIST TN 1297).
• Coverage: An expanded uncertainty with coverage factor $k$ gives an interval with a stated level of confidence (NIST TN 1297).

2. Working Principle

• Model: Write the measurand as a function of all inputs, including corrections (GUM).
• Evaluate inputs: Repeatability by Type A; resolution, calibration and environment by Type B with suitable distributions (NIST TN 1297).
• Combine: Apply the law of propagation, then Welch-Satterthwaite degrees of freedom (NIST TN 1297).
• Expand and report: Choose $k$, usually 2, and report the components (NIST TN 1297).

3. Applications

• Metrology tool qualification: Budgets such as the sheet-resistance example (Four-Point Probe page).
• Calibration certificates: Expanded uncertainties of reference standards (NIST TN 1297).
• Specification decisions: Guard bands at specification limits (Calibration and Traceability page).
• Process control: Check standards and control charts showing whether a measurement process is in statistical control (NIST TN 1297).

Evaluating Input Uncertainties

Name	Formula	Related Formulae	Variables/Constants
Type A standard uncertainty	$u\left(\bar{x}\right)=s\left(\bar{x}\right)=\sqrt{\frac{{\sum }_{k=1}^n{\left(x_k-\bar{x}\right)}^2}{n\left(n-1\right)}}=\frac{s}{\sqrt{n}}$	$\nu =n-1$	$x_k:repeatedobservations$ $\bar{x}:mean$ $s:experimentalstandarddeviation$ $n:numberofobservations$ $\begin{array}{r}\nu :degreesoffreedom;NISTTN1297:Eq.\\ A-5,standarddeviationofthemean,\nu =n\\ -1foraTypeAcomponent;GUM:TypeA\\ evaluationfromstatisticalanalysisof\\ seriesofobservations\end{array}$ $\begin{array}{r}(assumed:s=0.5percent,n=10;\\ computed:u=0.158percent)\end{array}$
Type B standard uncertainty	$u=\frac{a}{\sqrt{3}}\text{(rectangular)},u=\frac{a}{\sqrt{6}}\text{(triangular)},u^2=\frac{a^2\left(1+{\beta }^2\right)}{6}\text{(trapezoidal)}$	$u=1.48a\text{(normal, 50 per cent)},u=\frac{a}{3}\text{(normal, 99.73 per cent)}$	$a:half-widthofthelimits,(a_{+}-a_{-})/2$ $\begin{array}{r}\beta :ratiooftoptobasewidth;NISTTN\\ 1297:Eq.A-7withrectangular,triangular\\ andnormalcasesincluding1.48afor50\\ percentboundsandu=afor67percent;\\ GUM:trapezoidalEq.9areducingto\\ rectangularat\beta =1andtriangularat\beta =\\ 0\end{array}$ $\begin{array}{r}(computed:rectangular0.577a,triangular\\ 0.408a)\end{array}$
Reliability of a standard deviation	$\frac{\sigma \left(s\right)}{s}\approx \frac{1}{\sqrt{2\left(n-1\right)}}$	$n=10:24\text{per cent};n=50:10\text{per cent}$	$\begin{array}{r}\sigma (s)/s:relativeuncertaintyofthe\\ experimentalstandarddeviation;GUM:24\\ percentforn=10and10percentforn\\ =50(clause\mathrm{4.3.2});approximation\\ reproducingtheGUMvalues(computed)\end{array}$ $\begin{array}{r}(computed:23.6,16.2and10.1percentat\\ n=10,20and50)\end{array}$

Combining Uncertainties

Name	Formula	Related Formulae	Variables/Constants
Law of propagation of uncertainty	$u_c^2\left(y\right)={\sum }_{i=1}^N{c}_i^2{u}^2\left(x_i\right)+2{\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N{c}_i{c}_{j}u\left(x_i,x_j\right)$	$c_i=\frac{\partial f}{\partial x_i},u\left(x_i,x_j\right)=r_{ij}u\left(x_i\right)u\left(x_j\right)$	$y=f(x_1,...,x_N):measurandmodel$ $\begin{array}{r}{c}_i:sensitivitycoefficients,u(x_i,x_j):\\ covariance\end{array}$ $\begin{array}{r}{r}_{ij}:correlationcoefficient;NISTTN1297:\\ Eq.A-3includingcovarianceterms;GUM:\\ Eq.10andEq.11aforuncorrelatedinputs\end{array}$ $\begin{array}{r}(computed:twoequal0.1percent\\ componentsinseriescombineto0.141per\\ centuncorrelatedand0.2percentfully\\ correlated)\end{array}$
Relative form for products	${\left(\frac{u_c\left(y\right)}{y}\right)}^2={\sum }_i{\left(p_i\frac{u\left(x_i\right)}{x_i}\right)}^2$	$y=C{\prod }_i{x}_i^{p_i}$	$\begin{array}{r}{p}_i:exponentsofapower-lawmodel;GUM\\ clause\mathrm{5.1.6}:relativecombinedvariance\\ forpower-lawmodelswithexponentsof\\ negligibleuncertainty(formulagarbledin\\ extraction);Four-PointProbepage:R_s=\\ (\pi /\ln 2)(V/I)timesgeometriccorrections\end{array}$ $\begin{array}{r}(assumedfour-point-probebudget:\\ repeatability0.158,voltmeter0.1/\sqrt{3},\\ current0.05/\sqrt{3},geometry0.5/\sqrt{3},\\ temperature0.2/\sqrt{3}percent;computed:u_c=\\ 0.355percent,geometry66percentand\\ repeatability20percentofthevariance)\end{array}$
Welch-Satterthwaite degrees of freedom	${\nu }_{eff}=\frac{u_c^4\left(y\right)}{{\sum }_{i=1}^N\frac{c_i^4{u}^4\left(x_i\right)}{{\nu }_i}}$	${\nu }_i\to \infty \text{for well-known Type B limits}$	${\nu }_{eff}:effectivedegreesoffreedom$ $\begin{array}{r}{\nu }_i:degreesoffreedomofeachcomponent;\\ NISTTN1297:Eqs.B-1andB-2\end{array}$ $\begin{array}{r}(computed:{\nu }_{eff}=228forthe\\ four-point-probebudget,so{t}_{95}=1.97)\end{array}$

Expanded Uncertainty and Reporting

Name	Formula	Related Formulae	Variables/Constants
Expanded uncertainty	$U=k{u}_c\left(y\right)$	$Y=y\pm U$	$U:expandeduncertainty$ $\begin{array}{r}k:coveragefactor;NISTTN1297:k=2\\ givesabout95percent,k=3greater\\ than99percent,NISTconventionk=2\\ withothervaluesonlyfordocumented\\ requirements;GUM:factors1.64,1.96and\\ 2.58for90,95and99percent\end{array}$ $\begin{array}{r}(computed:normalquantilesk=1.000,\\ 1.645,1.960,2.000,2.576and3.000for\\ 68.27,90,95,95.45,99and99.73per\\ cent;U=0.71percentforthebudget)\end{array}$
Coverage factor from Student t	$k=t_p\left({\nu }_{eff}\right)$	$t_{95}:12.71(\nu =1),2.57(5),2.31(8),2.26(9),2.23(10),2.09(20),2.01(50)$	$\begin{array}{r}{t}_p:Studenttquantile;NISTTN1297:Table\\ B.1witht=2.31for\nu =8and2.26for\nu \\ =9,exampleofa100\mathrm{g}standardwith{u}_c=\\ 0.35\mathrm{mg}andk=2giving100.02147plus\\ orminus0.00070\mathrm{g};quantiles(computed\\ withSciPy)\end{array}$ $\begin{array}{r}(computed:2.26\times 0.35=0.79\mathrm{mg},matching\\ theTNexample)\end{array}$
Reporting	$\text{report }y,U,k\text{(or }{u}_c\text{), components and }\nu$	$\text{uncertainty to two significant digits}$	$\begin{array}{r}NISTTN1297:reportUwithkorreport{u}_c,\\ listcomponentswithdegreesoffreedom,\\ describeevaluation,explainkifnot2;\\ GUM:roundingexamplessuchasu(l)about\\ 0.06mm;two-digitconvention(myreading)\end{array}$ $\begin{array}{r}(myreading:listingthecomponentsletsa\\ readerrecomputeUforadifferentkor\\ seewhichtermtoimprove)\end{array}$

Important Questions

Ques: What is the difference between Type A and Type B uncertainty?

How the standard uncertainty is evaluated, not what kind of error it describes. Type A uses statistics of repeated observations: the standard uncertainty of a mean of $n$ readings is $s/\sqrt{n}$ with $n-1$ degrees of freedom (NIST TN 1297). Type B uses other knowledge, such as manufacturer limits or calibration certificates, converted to a standard uncertainty through an assumed distribution: $a/\sqrt{3}$ for a rectangular distribution of half-width $a$, $a/\sqrt{6}$ for triangular, and the GUM’s trapezoidal form between them. For 10 readings with an assumed 0.5 per cent standard deviation, the Type A uncertainty is 0.158 per cent (computed). Both types are combined in the same way, as variances. Reference: B. N. Taylor and C. E. Kuyatt, ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’, NIST Technical Note 1297 (January 1993) (fetch-tool summary of the OCR text); JCGM 100:2008, ‘Evaluation of measurement data - Guide to the expression of uncertainty in measurement’ (GUM), first edition 2008, corrected 2010 (fetch-tool summary of a partial text).

Ques: Does k = 2 give 95 per cent or 95.45 per cent coverage?

Both statements appear. NIST TN 1297 says that $k=2$ defines an interval with a level of confidence of about 95 per cent and adopts $k=2$ as its convention, while the GUM gives 1.96 as the factor for exactly 95 per cent with a normal distribution. For a normal distribution, $k=2$ covers 95.45 per cent and 95 per cent needs 1.960 (computed). Similarly, NIST says $k=3$ gives more than 99 per cent; the normal value is 99.73 per cent, and 99 per cent needs 2.576 (computed). The difference matters only with many degrees of freedom; with few, the t factor dominates. Reporting $k=2$ as approximately 95 per cent is therefore the more likely intended meaning (my reading). Reference: B. N. Taylor and C. E. Kuyatt, ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’, NIST Technical Note 1297 (January 1993) (fetch-tool summary of the OCR text); JCGM 100:2008, ‘Evaluation of measurement data - Guide to the expression of uncertainty in measurement’ (GUM), first edition 2008, corrected 2010 (fetch-tool summary of a partial text).

Ques: How do correlations change a combined uncertainty?

They add covariance terms to the sum of variances. The law of propagation is $u_c^2=\sum c_i^2{u}^2(x_i)+2\sum \sum c_i{c}_{j}u(x_i,x_j)$, where $c_i=\partial f/\partial x_i$ (NIST TN 1297; GUM). Two equal 0.1 per cent contributions combine to 0.141 per cent if independent but 0.2 per cent if fully correlated, for example two resistors calibrated against the same standard (computed). Negative correlation can reduce the result, which is why ratio measurements against a common reference are often more precise than either input alone (my reading). Ignoring correlation is a common source of underestimated budgets. Correlation coefficients can be estimated from simultaneous repeated readings (my reading), and NIST TN 1297 includes the covariance terms explicitly in its Eq. A-3, so they should never be dropped silently. Reference: B. N. Taylor and C. E. Kuyatt, ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’, NIST Technical Note 1297 (January 1993) (fetch-tool summary of the OCR text); JCGM 100:2008, ‘Evaluation of measurement data - Guide to the expression of uncertainty in measurement’ (GUM), first edition 2008, corrected 2010 (fetch-tool summary of a partial text).

Ques: What does a typical sheet-resistance uncertainty budget look like?

For an assumed four-point-probe measurement with repeatability of 0.5 per cent over 10 readings, voltmeter and current-source limits of 0.1 and 0.05 per cent, a geometric correction known to 0.5 per cent, and a temperature effect of 0.2 per cent, all limits treated as rectangular, the standard uncertainties are 0.158, 0.058, 0.029, 0.289 and 0.115 per cent (computed). Root-sum-square gives $u_c$ = 0.355 per cent; the geometric correction contributes 66 per cent of the variance and repeatability 20 per cent (computed). Welch-Satterthwaite gives 228 effective degrees of freedom, so $k$ = 1.97 for 95 per cent and $k$ = 2 is appropriate, giving $U$ = 0.71 per cent (computed; NIST TN 1297). Improving the probe geometry correction would help most (my reading). Reference: Four-Point Probe page of this vault; B. N. Taylor and C. E. Kuyatt, ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’, NIST Technical Note 1297 (January 1993) (fetch-tool summary of the OCR text).

Ques: When must the coverage factor exceed 2?

When the effective degrees of freedom are small. Welch-Satterthwaite gives ${\nu }_{eff}=u_c^4/\sum (c_i^4{u}_i^4/{\nu }_i)$ (NIST TN 1297), and $k$ is then the Student t quantile: 12.71 for one degree of freedom, 2.57 for 5, 2.31 for 8, 2.23 for 10 and 2.01 for 50 at 95 per cent (computed). NIST’s example of a 100 $\mathrm{g}$ standard with $u_c$ = 0.35 $\mathrm{mg}$ and 9 degrees of freedom uses 2.26, giving 0.79 $\mathrm{mg}$ (computed), although its reported certificate value uses $k=2$. The GUM notes that a standard deviation from 10 readings is itself uncertain by 24 per cent, and from 50 by 10 per cent (computed 23.6 and 10.1), which is why few readings need a larger factor. Reference: B. N. Taylor and C. E. Kuyatt, ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’, NIST Technical Note 1297 (January 1993) (fetch-tool summary of the OCR text); JCGM 100:2008, ‘Evaluation of measurement data - Guide to the expression of uncertainty in measurement’ (GUM), first edition 2008, corrected 2010 (fetch-tool summary of a partial text); Calibration and Traceability page of this vault.

References

1. B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, 1993. Available: https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote1297-1993.pdf
2. Joint Committee for Guides in Metrology, Evaluation of measurement data - Guide to the expression of uncertainty in measurement, JCGM 100:2008 (GUM 1995 with minor corrections). Available: https://users.physics.ox.ac.uk/~palmerc/Datafiles/BIPM.pdf
3. Four-Point Probe page of this vault.

Related pages in this vault: Physical Constants, Four-Point Probe, Process Corners and Statistical Models, Mismatch and Variability, Parameter Extraction Methods, Model Quality and Benchmark Tests, Spectroscopic Ellipsometer, Gauge Repeatability and Reproducibility (Gauge R&R), Calibration and Traceability, Statistical Process Control (SPC), Tool Matching and Total Measurement Uncertainty (TMU).