Start with the End in Mind

Global PFA computed via Drezner-Wesolowsky (1990) 20-point Gauss-Legendre quadrature of the bivariate normal CDF. Matches the workbook's L1/L2/L3 BVN engine. Specific risk at the acceptance limit shown separately on the Risk Calc tab.

Z540.3 establishes two independent sufficient conditions for global PFA ≤ 2 %: either TUR ≥ 4.6 : 1 (Method 6 guard band vanishes) or EOPR ≥ 89 % (Mimbs "Rule of 89"). Either dominance condition alone is sufficient.

Combined uncertainty: u_c = k × √((CMC/k)² + (resolution/3.464)² + repeatability²). Method 5 AL: AL = T − U·TUR·(1 − √(1 − 1/TUR²)). Method 6 AL: AL = T − U·max(0, 1.04 − exp(0.38·ln(TUR) − 0.54)). Test Limit: AL = T − U.

Inverse of the forward risk calculation: given a target global PFA, solve for the guard band that achieves it. Mirrors the workbook lambda PFA_GB_SOLVER - bisection on the guard-band factor g (acceptance limits become ±Tol·g) using the same accurate bivariate-normal risk engine the tables above use. If the PFA with no guard band is already at or below the target, the solver reports no guard band needed (g = 1).

Couples the risk engine to the cost model. Total expected cost as a function of the symmetric guard-band width w (per side, acceptance limits ±(Tol − w)) is C(w) = items · [cost/item + PFR(w)·retest cost + PFA(w)·false-accept cost], using the Production Inputs above. PFA(w) and PFR(w) are the same global probabilities the tables above show, from the accurate bivariate-normal engine (pfaWb / pfrWb) - there is no second risk math. The optimizer scans then golden-section refines over w in [0, Tol] to find the width that minimizes C(w). As w grows PFA falls and PFR rises, so the cost trades the two against each other.

The cost curve and optimum are exposed on window.__sp6CostCurve and window.__sp6Optimum; the optimum is validated against a brute-force scan and an independent scipy.optimize minimization to ≤ 1e-6 by EndInMind/validators/validate_cost_optimizer.py.

The forward risk engine above assumes the true value of the unit is normally distributed. For a skewed or one-sided process (e.g. a quantity bounded at zero), select a gamma process distribution instead. The measurement error stays normal (u_cal). This mirrors the workbook lambdas PFA_GAMMA / PFR_GAMMA / GB_WIDTH_GAMMA: the global PFA / PFR are integrated over the gamma true-value density with the same accurate normal acceptance probability the normal engine uses (no second normal-math path), and the gamma guard band is solved so PFA_GAMMA hits the target.

Edit equipment name, price, and 1σ uncertainty to model your own tiers. Default values reflect the seven Acme DMM tiers in the workbook.

Each line is one equipment tier. X-axis is years of production; Y-axis is cumulative cost including equipment, production, false rejects, and false accepts. Cheaper equipment with low TUR compounds risk cost every year.

n = ⌈ln(1 − Confidence) / ln(Reliability)⌉, from the binomial success-run theorem.

EOPR Definition

End-of-Period Reliability - probability that an instrument remains in tolerance at the end of its calibration interval.

Point Estimate

EOPR = 1 − c/n. Simple but ignores sample-size uncertainty.

Clopper-Pearson CI

Exact binomial confidence interval using the Beta distribution. Conservative (guaranteed coverage). NIST recommended.

Z540.3 "Rule of 89"

EOPR ≥ 89 % is the global dominance condition that bounds unconditional PFA ≤ 2 % regardless of TUR (Mimbs).

Assumptions

Bernoulli trial model - each calibration result is independent, probability of in-tolerance is constant across the population.

References

ANSI/NCSL Z540.3-2006 §B.2 • ILAC-P14:09/2020 • NASA-HDBK-8739.19 §4.3 • NCSLI RP-1 §5.4

What k means

k is the coverage factor used to expand a standard uncertainty u into an expanded uncertainty U = k · u. The choice of k sets the coverage probability - the fraction of the distribution captured by U around the measurand.

Common values

k = 1 → ≈ 68.27 % • k = 1.96 → 95.00 % • k = 2 → ≈ 95.45 % • k = 2.576 → 99.00 % • k = 3 → ≈ 99.73 %. Calibration certificates almost always quote U at k = 2 (about 95 % coverage).

k = 2 vs k = 1.96

k = 1.96 gives exactly 95.00 % under a normal distribution; k = 2 gives 95.45 %. GUM and ISO/IEC 17025 labs use k = 2 by convention because it is simple and conservative. ILAC G8:2019 §B.3 uses k = 1.96 for guard band sizing where exact 95 % matters.

When the distribution isn't normal

These tables assume a normal output distribution - the standard assumption when the combined uncertainty has enough degrees of freedom (Welch-Satterthwaite ν_eff ≳ 30). For low-DOF cases, use a t-distribution coverage factor instead.

Reference

JCGM 100:2008 (GUM) Annex G • NIST SP 811 §7.4 • ILAC G8:2019 §B.3 • ANSI/NCSL Z540.3 Handbook

Guard-banded rejection thresholds at the current confidence level and measurement uncertainty. Officer would not issue a ticket until the radar reads at least this value.

The analogy

An officer's radar gun reads your speed with some uncertainty. To be 95 % confident a driver is actually speeding before issuing a ticket, the officer must let the reading exceed the posted limit by enough margin that the true value almost certainly exceeds the limit too.

The formula

Threshold = Limit / (1 − u_m · NORMSINV(P)). With P = 95 %, z = 1.645 (single-sided). A 2 % relative uncertainty raises a 75 mph limit to ≈ 77.6 mph before a citation is justified.

Why this matters

ISO/IEC 17025:2017 §7.8.6 requires a documented decision rule when a statement of conformity is issued. Guarded rejection (ILAC G8:2019 Case 3) protects the customer from a false reject - mirror image of guarded acceptance protecting from a false accept.

Reference

ISO/IEC 17025:2017 §7.8.6 • ILAC G8:2019 §B (Cases 1-4) • JCGM 106:2012 §9

Example certificate language (Possible Pass): "Measured value: 0.82 units. Specification: ±1 unit. Decision rule: Specific risk with 2.5 % max PFA per side. Calculated specific PFA at this point: 3.4 %. Outcome: Possible Pass - within tolerance but specific risk exceeds the agreed threshold. Customer to determine disposition."

Rule of thumb: At shared-risk (no guard band), TUR ≳ 4.6 : 1 is a conservative boundary for PFA ≤ 2 %. Required U (k = 2) = Tolerance / 4.6. If EOPR ≳ 89 %, worst-case PFA ≤ 2 % even at lower TUR.

The "Measurement Stool" - three legs (Requirements, Equipment, Process) plus Verification and Continuous Improvement.

• ISO/IEC 17025:2017 - Primary standard for lab competence. Clauses 3.7, 7.1.3, 7.8.6.1, 7.8.6.2 govern decision rules.
• UKAS LAB 48 (Ed. 5, 2024) - Practical guidance with worked examples. Highly recommended reading.
• ILAC-G8:09/2019 - International guidelines on decision rules. Covers guard band calculation, PFA, and reporting.
• JCGM 106:2012 - Statistical foundation for understanding risk in conformity assessment.
• ANSI/NCSL Z540.3 Handbook - The 2 % PFA rule and practical methods (Methods 5 and 6).
• ASME B89.7.3.1-2001 - Guidelines for decision rules considering measurement uncertainty.
• ASME B89.7.4.1-2005 - Measurement uncertainty and conformance testing: risk analysis.
• ISO 14253-1:2017 - Decision rules for verifying conformity or nonconformity with specifications.
• NIST SP 811 - Guide for the Use of the International System of Units (SI).

ISO/IEC 17025:2017 §6.5.1: The laboratory shall establish and maintain metrological traceability of its measurement results by means of a documented unbroken chain of calibrations, each contributing to the measurement uncertainty, linking them to an appropriate reference.

Definition (VIM / ISO Guide 99): Metrological Traceability - property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to measurement uncertainty.

Cumulative effect: Uncertainty is cumulative from one level of the hierarchy to the next. Lowering the reference CMC by ~10× can transform a 6 %+ PFA failure into a near-zero PFA result on the same instrument.

Measurement Uncertainty (definition): The doubt that exists about a measurement's result. Every measurement - even the most careful - carries some uncertainty.

CMC - Calibration and Measurement Capability. Typical CMC standard uncertainty contributors:

• Repeatability
• Resolution
• Reproducibility
• Reference Standard Uncertainty
• Reference Standard Stability
• Environmental Effects
• Operator / Method Effects

Same root cause: both errors come from uncertainty overlapping the specification limit. Larger uncertainty = more of both. The consequences are not symmetric - consumer risk can mean loss of life or mission, while producer risk typically means unnecessary rework. Both carry cost; consumer risk has the greater potential for major negative consequences and must be tightly controlled.

The football analogy. Specific Risk = "Did this kick clear these posts?" (one ball, one outcome). Global Risk = "What is the kicker's career field-goal percentage from this distance?" (the long-run frequency).

Where 4:1 came from: Hayes & Crandon (U.S. Navy, 1955) proposed 4:1 as a slide-rule-era compromise targeting 1 % consumer risk under specific assumptions. It is not a universal law.

ANSI/NCSL Z540.3: TUR = (USL − LSL) / (2 · U₉₅) for the calibration process - ratio of the tolerance span to twice the expanded uncertainty.

TUR vs TAR - not the same. TAR uses only the manufacturer's accuracy spec for the standard. A 25:1 TAR for a digital micrometer can become a much lower TUR once resolution, repeatability, and environmental effects are included.

Bench-level dominance: at TUR ≥ 4.6 : 1, Method 6 yields M ≤ 0 and the guard band vanishes - the full tolerance is usable. Required U for a given tolerance (PFA ≤ 2 %): U_exp (k = 2) ≈ Tolerance / 4.6.

Plain definition: EOPR = number of calibrations meeting acceptance criteria ÷ total number of calibrations. It is the population-level success rate over a calibration interval.

Sample size formula: Sample Size = ln(1 − Confidence) / ln(Target Reliability). For 95 % EOPR at 95 % confidence with 0 failures, n = 59.

Global dominance: EOPR ≥ 89 % is sufficient to bound unconditional PFA (UFAR) at ≤ 2 % regardless of TUR (Mimbs - Using Reliability to Meet Z540.3's 2 % Rule).

Take-away from Morehouse load-cell data: achieving better than 0.02 % of applied force typically requires very high-end equipment (meters + load cells together).

Star Wars - Specific Risk Acceptance Limit

Setup: Vent port 2 m wide, proton torpedo 0.5 m, standard uncertainty u = 0.125 (k = 1), applied as U at k = 2 → U = 0.25.

Math: GB multiplier = NORM.S.INV(0.975) / 2 = 0.980. Guard band = 0.980 · (2 · 0.125) = 0.245. Acceptance interval narrows to ±0.755 m.

Lesson: Even when the system looks comfortable on paper, uncertainty consumes 24.5 % of each half-tolerance.

Radar Gun - Global vs Specific Risk

Setup: Speed-limit zone 60 mph ± 3 mph. Two radar guns at points A and B; TUR = 6:1.

Specific Risk view: Car clocks 65 mph at A and 55 mph at B - each point individually outside the limit. Specific risk treats every measurement on its own.

Global Risk view: Across the segment, the car's average speed is 60 mph and 98 % of 10 000 sampled cars are between 57 and 63 mph - the population-level risk is well-bounded.

Resistor Verification

Setup: Fixed resistor 1500 Ω ± 0.2 Ω (±0.013 %). DMM at TUR ~1.44 : 1.

Production reality: ~30 % flagged failing the global GB. Retest in the metrology lab shows < 0.2 % were actual failures - the rest were producer-risk false rejects.

Fix: A more accurate DMM at ~3× the price gives expected false-reject rate 1.035 % and consumer risk < 1 %.

Accuracy = low bias (results center on true value). Precision = low random error (results cluster tightly).

The standard 4:1 assumption. The classic TUR/TAR rules assume the measurement process is centered (bias corrected). Uncorrected bias dramatically increases PFA on one side of the distribution.

Worked example - force calibration with +9 lbf bias: An indicator reads 10 000 lbf when the true applied force is 10 009 lbf (bias = +9 lbf, TUR = 58.75:1). Without correction the producer-risk PFR is enormous and many in-tolerance instruments are scrapped.

The Deming Funnel. Don't adjust without understanding. Constant ad-hoc adjustments without understanding the process never hit the target. The right response is to characterize the bias, then correct it consistently.

Bottom line: not correcting for measurement bias is a common problem - customers may be receiving calibrations that are far worse than the certificate suggests.

The spec: NFL rulebook ball pressure 12.5 - 13.5 psi. Tolerance ±0.5 psi (half-tolerance).

The instrument actually used: Two gauges - one "no name," one Wilson CJ-01 (made by Jiao Hsiung Industry Corp.). Wilson states no measurement uncertainty.

TUR reality: At best the Wilson provides ±3.3 psig uncertainty (~0.817 × 4 with 4:1 desired) - that is 6.6× less accurate than required.

The fix: An Additel GP30 (0.001 psig resolution, ±0.05 % FS, calibration U ≈ ±0.003 psig at k = 2) costs ~$714 including accredited calibration.

The price of getting it wrong: Deflate Gate totalled more than $22.5 M in investigation cost. The NFL used a $30 gauge in place of a $700 instrument capable of doing the job.

The lens: Total cost = Equipment + Production + (False Reject × retest cost) + (False Accept × downstream cost).

Two questions before any measurement or purchase: (1) How good does the measurement need to be? (business lens - consequences, regulatory exposure). (2) How will you achieve and verify that capability? (technical lens - equipment, process, reliability evidence).

1. Calculating Measurement Uncertainty correctly is essential to everything that follows, including decision rules.
2. Metrological Traceability relies on an unbroken chain of calibrations, each contributing to measurement uncertainty.
3. A decision rule describes how measurement uncertainty is accounted for when stating conformity (ISO/IEC 17025 §3.7, §7.1.3).
4. Not correcting for known bias increases measurement risk and undermines traceability.
5. The 4:1 TUR rule provides a specific risk level only under explicit assumptions - it is not a universal law.
6. Choose specific risk when per-point control is critical (aerospace, medical, safety).
7. Choose global risk for high-volume or routine calibration where population-level control is sufficient.
8. TUR ≥ 4.6 : 1 limits PFA to < 2 % under global risk models; EOPR ≥ 89 % is the global counterpart.
9. Guard banding is a business decision - balance consumer risk vs producer cost based on consequences.
10. Lower measurement uncertainty (better lab) directly reduces both PFA and PFR - the best risk mitigation.
11. Decision rules are not optional - ISO/IEC 17025:2017 requires that measurement uncertainty be accounted for.

Standards & Guidance Documents

• ILAC G8:09/2019 - Guidelines on Decision Rules and Statements of Conformity
• JCGM 106:2012 - Evaluation of measurement data: The role of measurement uncertainty in conformity assessment
• UKAS LAB 48 - Decision Rules and Statements of Conformity
• ISO/IEC 17025:2017 - General requirements for the competence of testing and calibration laboratories
• Handbook for the Application of ANSI/NCSL Z540.3-2006 - Requirements for the Calibration of M&TE
• The Metrology Handbook, 3rd Edition - Chapter 30
• NCSLI RP-18 - Estimation and Evaluation of Measurement Decision Risk
• ASME B89.7.3.1-2001 - Guidelines for Decision Rules
• ASME B89.7.4.1-2005 - Measurement Uncertainty and Conformance Testing: Risk Analysis
• ISO 14253-1:2017 - Decision rules for proving conformity or nonconformity with specifications
• WADA Technical Document - TD2017DK
• Decision Rule Guidance v1.41 - Cenker, Zumbrun, Shah, et al.

Key Papers

• Delker, C. - Evaluation of Guard Banding Methods for Calibration and Product Acceptance
• Deaver, D. & Sompri, J. - A Study of and Recommendations for Applying the False Acceptance Risk Specification of Z540.3
• Rishi, S. - Guard-banding Methods: An Overview
• Dobbert, M. - A Guard-Band Strategy for Managing False-Accept Risk
• Dobbert, M. - Understanding Measurement Risk
• Harben, J. & Reese, P. - Risk Mitigation Strategies for Compliance Testing
• Mimbs, S. - Measurement Decision Risk: The Importance of Definitions
• Mimbs, S. - Conformance Testing: Measurement Decision Rules
• Mimbs, S. - Using Reliability to Meet Z540.3's 2 % Rule (Rule of 89)
• Castrup, H. - Analytical Metrology SPC Methods for ATE Implementation
• Zumbrun, H. & Cenker, G. - The Force of Decision Rules: Applying Specific and Global Risk to Star Wars
• Cenker, G. & Zumbrun, H. - Unraveling the Tom Brady Deflate Gate
• Reese, P. - Calibration in Regulated Industries: Federal Agency use of ANSI Z540.3 and ISO 17025

More from Morehouse & IndySoft

• Morehouse Instrument Company - www.mhforce.com • sales@mhforce.com • (717) 843-0081
• Morehouse NCSLI Force Course - mhforce.com/ncsli-force-course/
• Morehouse Force ILC / PT - mhforce.com/calibration/force-ilc-and-pt
• IndySoft Calibration Management Software - www.indysoft.com • greg.cenker@indysoft.com

• Environmental conditions block on method cover sheet - Section 4
• Personnel competency & training records reference - Section 4
• Technical review and approval signatures pending
• Internal audit schedule entry per ISO/IEC 17025:2017 Cl. 8.8 - target frequency: annual

Halton quasi-Monte Carlo PFA tightens toward the exact value as n grows, mirroring Henry's Halton columns in the V14 workbook.

Pc = P(measured value conforming | true value at upper tolerance) = Φ(1) − Φ(−9), exact = 0.841344746068543.

Derived ratios from the resistor example parameters. Cm = (T_U−T_L) / (4·u_meas); Cp = (T_U−T_L) / (6·σ_proc); guard-band ratio r = (T_U−A_U) / (2·u_meas).

The solver finds the guard band that drives global PFA to a target, mirroring the workbook lambda PFA_GB_SOLVER (bisection on the guard-band factor). Verification: feeding the solved acceptance limits back through the app's accurate risk engine reproduces the target PFA, and the solved factor matches an independent bisection. Inputs below use the JCGM resistor process/measurement σ (0.12 / 0.04) and tolerance ±0.2.

A target at or above the no-guard-band PFA (1.894 %) returns g = 1 (no guard band needed) - the last row exercises that clamp. For the symmetric global-PFA case any positive target below that is feasible (collapsing the acceptance window drives PFA toward zero); the infeasible branch guards the one-sided case where a single tail has an irreducible minimum PFA.

Global PFA computed three independent ways for whatever inputs are currently set on the Risk Calc tab. Pairwise deltas confirm agreement.

Inputs: −

This web version is a port of the V10 workbook with all calculations re-implemented in JavaScript. No data is transmitted; all computation happens client-side. The reference workbook remains the authoritative source for audit and certification.

Statistical primitives

Calculation differences vs V10 workbook

• None on calculated values. Every visible output matches the workbook to <= 1e-3 relative error at every tested input.
• Display precision is higher in some cells. Methods U₉₅ shows 5 decimal places (workbook 4); TUR shows 4 decimal places (workbook 2); Cost Model PFA/PFR show 5 decimal places on percent (workbook 3); bias-correction PFA shows 6 decimal places (workbook 2). The extra digits ensure CSV exports round-trip without precision loss.
• L1 uncertainty budget is exposed in the UI. The workbook has these components in hidden cells F77:F82 on the Risk Pyramid sheet. The web app surfaces them as 5 editable inputs plus an auto-derived Ref Std term from the L2 tolerance.
• L2 / L3 tolerance convention preserved exactly. The workbook treats the L1 tolerance input as half-width fraction (T/2 of NV) but L2/L3 as full-width fraction (T of NV, halved internally per cell B91). The web app mirrors this asymmetry.

Features added in the web port (not in workbook)

• Per-tab CSV export - each calc tab can dump its inputs, summary, and all result tables to a single sectioned CSV.
• CSV import for Methods readings (10x5 grid) and Cost Model equipment roster. Round-trip safe: an exported CSV re-imports cleanly.
• Chart PNG download on the Risk Curve and Cost Model charts.
• Session save / load. Save all inputs to a JSON file or restore from one. Continuous auto-save to localStorage prevents loss on accidental tab close.
• Mobile-responsive. Sidebar collapses to hamburger under 900 px; tables scroll horizontally inside their container; works down to 360 px width.
• Single-file deploy. No build step; CDN dependencies only.

Calculation paths that are bit-for-bit ports from the workbook

• All 7 guard-band methods (Simple Acceptance, Specific Risk, ILAC G8, Dobbert M6, RSS/Method 5, Fixed 0.98U, k=2 Subtraction).
• Risk Pyramid cascade conformance (1 - PFA_L1) x (1 - PFA_L2) x (1 - PFA_L3) using the workbook's L1 budget RSS combination.
• Cost Model PFA / PFR with Dobbert M6 guard band applied before BVN integration (matches workbook LAMBDA PFA(-T, T, -T+gb, T-gb, 0, sigt, sigp)).
• Methods TUR = T_half / (2 * u_std), using k=2 expanded uncertainty in the denominator regardless of user's input k.
• Clopper-Pearson EOPR bounds and sample-size formulas for Reliability.
• Two-way coverage factor (k <-> p) lookup tables.
• Guarded Rejection threshold and reference tables.
• Bias correction worked-example: Known Bias = |NV - x_m|, DUT Reading = NV + 0.8*T_half, NORM.DIST PFA before and after subtracting bias.
• Asymmetric tolerance dispatching via the workbook's IS_SYMMETRIC(tl, tu, al, au, nom) rule with separate PFA_ASYMMETRIC / PFR_ASYMMETRIC branches.

Validation evidence

Validated cell-by-cell against the V10 workbook driven by Excel via COM automation. 531 individual numeric comparisons across 5 input scenarios (workbook defaults plus 4 varied scenarios spanning TUR 0.5:1 to 167:1, ITP 0.85 to 0.999, both symmetric and asymmetric tolerances), all matching to <= 1e-3 relative error or the floating-point noise floor of the 6-point BVN quadrature. Full validation suite (validators/) and run instructions are bundled with the deliverable in the EndInMind folder.

Enter the indicator response for each run at each applied force. Responses are taken as already zero-corrected (the value used in the fit). Use Ascending + Descending to add a returning sweep for hysteresis. The fit, S, LLF and class limits are computed from every populated per-run point via e74Fit.

CSV import format (header auto-detected): dir,force,run1,run2,… — dir is A (ascending) or D (descending); omit it for ascending. Reuses the app's CSV importer.

Observed response − fitted response at each per-run point, in output units. Tight, randomly scattered residuals indicate a good fit; structure (curvature, fanning) suggests a higher order or an instrument effect.

The 33-point NIST LOADCELL.DAT example (11 loads × 3 repetitions, ASTM V8.xlsx Validation sheet) fit at order 2 by window.__e74SelfTest(), compared to the published NIST reference coefficients and S. PASS when within the Validation-sheet tolerances. This runs in-browser, offline, every time the tab is shown.