Guard band and global PFA / PFR
Global false-accept and false-reject risk for a calibration process, from its TUR and the in-tolerance rate of the instruments you receive (EOPR). Evaluate any guard band, or solve numerically for the smallest one that keeps PFA at or below 2 %. Global risk describes the process, not one reading.
Guard band · global risk
v0.1.0 · JCGM 106 · Z540.3Guard band neededw = 0.05369 × UAcceptance limits at ±97.32 % of the tolerance. Solved numerically from the PFA model, not an approximation.
Global PFA · PFR2.0 % · 5.9 %With simple acceptance: PFA 2.3 % · PFR 5.1 %. Global risk is for the process, not one reading.
Wording for a procedure or certificate
Decision rule: guarded acceptance, w = 0.0536935 × U (acceptance limits ±97.3153 % of the tolerance). Global risk for this calibration process (over the population of instruments, not for an individual result): PFA 2.0 %, PFR 5.9 %, for TUR 2:1, EOPR 90 %, k = 2, normal distributions (JCGM 106:2012; ANSI/NCSL Z540.3 Handbook method).
PFA = P(|e_uut| > T, |e_uut + e_m| ≤ T − w) · σ_uut = T / Φ⁻¹((1 + EOPR)/2) · U = T / TUR
Test vectors for this tool
The reference cases this calculator is tested against on every change. Use them to check your own software. All test vectors · Download JSON
| ID | Inputs | Expected | Tolerance | Source |
|---|---|---|---|---|
| GBR-01 | tur = 4, eopr = 0.95, k = 2, multiplier = 0 | 0.00858266480892313 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-02 | tur = 4, eopr = 0.95, k = 2, multiplier = 0 | 0.01553651303063799 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-03 | tur = 4, eopr = 0.95, k = 2, multiplier = 1 | 0.00020770265389761123 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-04 | tur = 4, eopr = 0.95, k = 2, multiplier = 1 | 0.1035719259168547 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-05 | tur = 1, eopr = 0.8, k = 2, multiplier = 0 | 0.05419268258331656 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-06 | tur = 2, eopr = 0.9, k = 2, multiplier = 0 | 0.02263849861835233 probability | ± 1e-12 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. |
| GBR-07 | tur = 2, eopr = 0.9, k = 2, targetPfa = 0.02 | 0.05369353322830693 × U | ± 1e-9 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. Root of PFA(r) = 0.02 found with mpmath findroot. |
| GBR-08 | tur = 1.5, eopr = 0.85, k = 2, targetPfa = 0.02 | 0.2287841078663865 × U | ± 1e-9 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. Root of PFA(r) = 0.02 found with mpmath findroot. |
| GBR-09 | tur = 10, eopr = 0.99, k = 2, targetPfa = 0.02 | 0 × U | ± 0 | Independent evaluation of the JCGM 106:2012 bivariate-normal global-risk model with mpmath 1.4.1 at 30 significant digits, by two formulations (integrating over the UUT error and over the measurement error) that agree to better than 1e-30. Not yet checked against a published table: the ILAC, BIPM, NASA and NCSLI documents could not be opened from the build environment. Simple acceptance PFA is about 0.12 %, below the target. |
| GBR-10 | tur = 4, eopr = 1, k = 2, multiplier = 0 | OutOfRangeError | exact | Metrology Bench rule: refuse rather than extrapolate (EOPR accepted from 0.5 to 0.9999) |
| GBR-11 | tur = 0.2, eopr = 0.95, k = 2, multiplier = 0 | OutOfRangeError | exact | Metrology Bench rule: refuse rather than extrapolate (TUR accepted from 0.5 to 100) |
| GBR-12 | tur = 2.5, eopr = 0.9044192954543706, k = 2, multiplier = 0.25 | 0.01 probability | ± 0.005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.3 Example (precision resistors), pp. 27–28 (PDF pp. 36–37): printed RC = 0.01 = 1 %. Inputs: T = ±0.2 Ω, u0 = 0.12 Ω, um = 0.04 Ω, w = 0.02 Ω = 0.25U. |
| GBR-13 | tur = 2.5, eopr = 0.9044192954543706, k = 2, multiplier = 0.25 | 0.07 probability | ± 0.005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.3 Example (precision resistors), p. 28 (PDF p. 37): printed RP = 0.07 = 7 %. |
| GBR-14 | tur = 2, eopr = 0.9973002039367398, k = 2, multiplier = 0 | 0.001 probability | ± 0.0005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.6.2 and Figure 17, p. 33 (PDF p. 42): u0 = T/6, Cm = T/(4um) = 2, w = 0 gives RC ≈ 0.1 % (one significant figure, read from Figure 17). |
| GBR-15 | tur = 2, eopr = 0.9973002039367398, k = 2, multiplier = 0 | 0.015 probability | ± 0.0005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.6.2 and Figure 17, p. 33 (PDF p. 42): RP ≈ 1.5 % for Cm = 2, w = 0 (read from Figure 17). |
| GBR-16 | tur = 10, eopr = 0.9973002039367398, k = 2, multiplier = 0 | 0.0004 probability | ± 0.00005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.6.3 and Figure 17, p. 33 (PDF p. 42): Cm = 10 gives RC ≈ 0.04 % (read from Figure 17). |
| GBR-17 | tur = 10, eopr = 0.9973002039367398, k = 2, multiplier = 0 | 0.0007 probability | ± 0.00005 | JCGM 106:2012 (BIPM, https://www.bipm.org/documents/20126/2071204/JCGM_106_2012_E.pdf), 9.5.6.3 and Figure 17, p. 33 (PDF p. 42): Cm = 10 gives RP ≈ 0.07 % (read from Figure 17). |
| GBR-18 | tur = 97.9981992270027, eopr = 0.95, k = 2, multiplier = 1 | 0.002406 probability | ± 5e-7 | NASA-HDBK-8739.19-4 (Baseline 2010-07-13, revalidated 2018-03-01, standards.nasa.gov), 5.3.2.2 Table 5-2 (k-factor 2, UFAR option), p. 60 (PDF p. 80): uncertainty ratio 100:1, printed false reject risk 0.2406 %. |
| GBR-19 | tur = 48.99909961350135, eopr = 0.95, k = 2, multiplier = 1 | 0.004931 probability | ± 5e-7 | NASA-HDBK-8739.19-4 (Baseline 2010-07-13, revalidated 2018-03-01, standards.nasa.gov), 5.3.2.2 Table 5-2 (k-factor 2, UFAR option), p. 60 (PDF p. 80): uncertainty ratio 50:1, printed false reject risk 0.4931 %. |
| GBR-20 | tur = 4.899909961350135, eopr = 0.95, k = 2, multiplier = 1 | 0.076273 probability | ± 5e-7 | NASA-HDBK-8739.19-4 (Baseline 2010-07-13, revalidated 2018-03-01, standards.nasa.gov), 5.3.2.2 Table 5-2 (k-factor 2, UFAR option), p. 60 (PDF p. 80): uncertainty ratio 5:1, printed false reject risk 7.6273 %. |
| GBR-21 | tur = 1.1, eopr = 0.5715, k = 1.96, multiplier = 0 | 0.06956 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.1:1, in-tolerance probability for maximum false-accept risk 57.15 %, printed maximum false-accept risk 6.956 % (TUR 1.1 also in Table 1, p. 8, where listed). |
| GBR-22 | tur = 1.1, eopr = 0.5715, k = 1.96, targetPfa = 0.02 | 0.4368 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.1:1, printed M × 100 % = 43.68 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-23 | tur = 1.2, eopr = 0.5789, k = 1.96, multiplier = 0 | 0.06495 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.2:1, in-tolerance probability for maximum false-accept risk 57.89 %, printed maximum false-accept risk 6.495 % (TUR 1.2 also in Table 1, p. 8, where listed). |
| GBR-24 | tur = 1.2, eopr = 0.5789, k = 1.96, targetPfa = 0.02 | 0.4158 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.2:1, printed M × 100 % = 41.58 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-25 | tur = 1.3, eopr = 0.5854, k = 1.96, multiplier = 0 | 0.06092 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.3:1, in-tolerance probability for maximum false-accept risk 58.54 %, printed maximum false-accept risk 6.092 % (TUR 1.3 also in Table 1, p. 8, where listed). |
| GBR-26 | tur = 1.3, eopr = 0.5854, k = 1.96, targetPfa = 0.02 | 0.3959 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.3:1, printed M × 100 % = 39.59 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-27 | tur = 1.5, eopr = 0.5962, k = 1.96, multiplier = 0 | 0.0542 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.5:1, in-tolerance probability for maximum false-accept risk 59.62 %, printed maximum false-accept risk 5.420 % (TUR 1.5 also in Table 1, p. 8, where listed). |
| GBR-28 | tur = 1.5, eopr = 0.5962, k = 1.96, targetPfa = 0.02 | 0.3589 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.5:1, printed M × 100 % = 35.89 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-29 | tur = 1.75, eopr = 0.6067, k = 1.96, multiplier = 0 | 0.04763 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.75:1, in-tolerance probability for maximum false-accept risk 60.67 %, printed maximum false-accept risk 4.763 % (TUR 1.75 also in Table 1, p. 8, where listed). |
| GBR-30 | tur = 1.75, eopr = 0.6067, k = 1.96, targetPfa = 0.02 | 0.3172 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 1.75:1, printed M × 100 % = 31.72 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-31 | tur = 2, eopr = 0.615, k = 1.96, multiplier = 0 | 0.04249 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 2:1, in-tolerance probability for maximum false-accept risk 61.50 %, printed maximum false-accept risk 4.249 % (TUR 2 also in Table 1, p. 8, where listed). |
| GBR-32 | tur = 2, eopr = 0.615, k = 1.96, targetPfa = 0.02 | 0.2793 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 2:1, printed M × 100 % = 27.93 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-33 | tur = 2.5, eopr = 0.6271, k = 1.96, multiplier = 0 | 0.03495 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 2.5:1, in-tolerance probability for maximum false-accept risk 62.71 %, printed maximum false-accept risk 3.495 % (TUR 2.5 also in Table 1, p. 8, where listed). |
| GBR-34 | tur = 2.5, eopr = 0.6271, k = 1.96, targetPfa = 0.02 | 0.2122 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 2.5:1, printed M × 100 % = 21.22 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-35 | tur = 3, eopr = 0.6355, k = 1.96, multiplier = 0 | 0.02968 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 3:1, in-tolerance probability for maximum false-accept risk 63.55 %, printed maximum false-accept risk 2.968 % (TUR 3 also in Table 1, p. 8, where listed). |
| GBR-36 | tur = 3, eopr = 0.6355, k = 1.96, targetPfa = 0.02 | 0.1536 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 3:1, printed M × 100 % = 15.36 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-37 | tur = 3.5, eopr = 0.6418, k = 1.96, multiplier = 0 | 0.02579 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 3.5:1, in-tolerance probability for maximum false-accept risk 64.18 %, printed maximum false-accept risk 2.579 % (TUR 3.5 also in Table 1, p. 8, where listed). |
| GBR-38 | tur = 3.5, eopr = 0.6418, k = 1.96, targetPfa = 0.02 | 0.1011 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 3.5:1, printed M × 100 % = 10.11 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-39 | tur = 4, eopr = 0.6465, k = 1.96, multiplier = 0 | 0.02281 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 4:1, in-tolerance probability for maximum false-accept risk 64.65 %, printed maximum false-accept risk 2.281 % (TUR 4 also in Table 1, p. 8, where listed). |
| GBR-40 | tur = 4, eopr = 0.6465, k = 1.96, targetPfa = 0.02 | 0.0532 × U | ± 0.00005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 4:1, printed M × 100 % = 5.32 % (2 % false-accept risk, acceptance limit A = T − U95·M, Equation 3, p. 7). |
| GBR-41 | tur = 5, eopr = 0.6534, k = 1.96, multiplier = 0 | 0.01852 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 5:1, in-tolerance probability for maximum false-accept risk 65.34 %, printed maximum false-accept risk 1.852 % (TUR 5 also in Table 1, p. 8, where listed). |
| GBR-42 | tur = 6, eopr = 0.658, k = 1.96, multiplier = 0 | 0.01559 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 6:1, in-tolerance probability for maximum false-accept risk 65.80 %, printed maximum false-accept risk 1.559 % (TUR 6 also in Table 1, p. 8, where listed). |
| GBR-43 | tur = 8, eopr = 0.664, k = 1.96, multiplier = 0 | 0.01184 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 8:1, in-tolerance probability for maximum false-accept risk 66.40 %, printed maximum false-accept risk 1.184 % (TUR 8 also in Table 1, p. 8, where listed). |
| GBR-44 | tur = 10, eopr = 0.6676, k = 1.96, multiplier = 0 | 0.00955 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 10:1, in-tolerance probability for maximum false-accept risk 66.76 %, printed maximum false-accept risk 0.955 % (TUR 10 also in Table 1, p. 8, where listed). |
| GBR-45 | tur = 12, eopr = 0.6701, k = 1.96, multiplier = 0 | 0.008 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 12:1, in-tolerance probability for maximum false-accept risk 67.01 %, printed maximum false-accept risk 0.800 % (TUR 12 also in Table 1, p. 8, where listed). |
| GBR-46 | tur = 15, eopr = 0.6726, k = 1.96, multiplier = 0 | 0.00643 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 15:1, in-tolerance probability for maximum false-accept risk 67.26 %, printed maximum false-accept risk 0.643 % (TUR 15 also in Table 1, p. 8, where listed). |
| GBR-47 | tur = 19, eopr = 0.6747, k = 1.96, multiplier = 0 | 0.0051 probability | ± 0.000005 | M. Dobbert, "A Guard-Band Strategy for Managing False-Accept Risk", NCSLI Workshop & Symposium 2008, Keysight white paper 5991-1267EN (Aug 2023 printing), Table 2, p. 11: TUR 19:1, in-tolerance probability for maximum false-accept risk 67.47 %, printed maximum false-accept risk 0.510 % (TUR 19 also in Table 1, p. 8, where listed). |
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