Where One Term Ends and the Next Begins, Part 4 of 7: Test Uncertainty Ratio (TUR) - A Screen, Not a Decision Rule

Figure 7. Consumer risk against accuracy ratio. The curve passes through roughly 1 % consumer risk near a 3:1 ratio, the relationship Hayes worked on.

Jerry Hayes of the United States Navy had a problem in the mid-1950s. Measurement decision risk had been worked out on paper by Eagle, Grubbs, and Coon, and almost nobody could use it. There was no computing power to speak of. So, Hayes accepted a consumer's risk of about 1 % and worked out that roughly a 3:1 accuracy ratio would deliver it. Assuming a 95 % confidence level for each, Hayes used specifications for the test ratio: the numerator was the unit under test's specifications, and the denominator the calibration standards' specifications. The intent was to use procedure controls to mitigate other measurement errors during testing. Not fully trusting that all specifications were equal, Hayes and Crandon increased the ratio to 4:1. Three became four. The Navy adopted 4:1, and much of the metrology community followed. The 4:1 rule therefore provides a specific level of risk only under explicit assumptions and conditions.

That rule was always meant as a placeholder until the math got easier. The math got easier decades ago. Yet, the placeholder is still on purchase orders, and some argue to keep the 4:1 TAR rule. TAR, Test Accuracy Ratio, is not a typo for TUR: the original rule compared accuracy specifications, not uncertainties, because specifications were what was simple to use at the time. In some fairness, many manufacturers of metrology equipment lived on reputation in those days, so their specifications were taken at face value.

What TUR Is

The Test Uncertainty Ratio is the tolerance of the item under test divided by the expanded uncertainty of the calibration process. ILAC-G8 writes it as TUR = TL/U. ANSI/NCSL Z540.3 writes the same thing with the span of a symmetric tolerance over twice the expanded uncertainty. JCGM 106 and UKAS LAB 48 call it the measurement capability index, C_m, and LAB 48 defines C₉₅ = (TU − TL)/(2·U₉₅). For a symmetric tolerance, every one of those gives the same number.

The denominator is where labs go wrong. It is not the reference standard's accuracy; that was the old Test Accuracy Ratio, and it ignored everything else. It is the whole calibration process uncertainty: the reference's CMC, the unit under test's resolution and repeatability, the environment, and the other contributors, combined in quadrature, excluding only the unit's own bias. ILAC-P14 asks for the same contents on the certificate. If your TUR denominator is smaller than your certificate uncertainty, you have calculated a TAR and called it a TUR.

What TUR Does

TUR answers one question: is this measurement process capable enough to make the decision at all? It is a screening ratio, and it screens at the population level. Dobbert's managed guardband work, and the Reese and Harben paper that followed it, showed that at a TUR of 4.6:1 or better the worst-case global false accept risk stays below 2 % with no guardband at all. That is useful. It is what UKAS LAB 48 means when it says the role of uncertainty under simple acceptance is to act as a precondition: PASS when the value is within tolerance AND C₉₅ ≥ 5, FAIL otherwise.

What TUR Does Not Do

It does not change the risk of the one decision in front of you. Picture the measurement result as a bell curve centered on the observed value, with its width set by the uncertainty. Move that curve so its center sits exactly on the tolerance limit. Half the area is outside. The specific probability that this item is actually out of tolerance is about 50 %, and it is 50 % at a TUR of 2:1, at 4:1, and at 10:1. A higher TUR narrows the curve, so fewer results land close enough to the limit for this to matter, and the guardband needed to control the risk gets smaller. But at the limit, the coin flip is the coin flip.

Figure 8. The 50 % problem. Under simple acceptance, a measured value exactly at the tolerance limit has half its distribution outside the specification, whatever the TUR.

That coin flip has a name: simple acceptance, often called shared risk. The measured value is reported, no guardband is applied (w = 0), and anything at or inside the tolerance passes. ISO/IEC 17025 permits it only when the rule is explicitly defined and agreed with the customer, and the customer accepts a probability of false acceptance that can reach 50 % at the tolerance limit. The standard does not prescribe an acceptable risk level; it requires that the risk be understood and documented.

Figure 9. Same instrument, same tolerance, same measured value of 10 008 lbf. Only the calibration source changes, and with it the uncertainty, the TUR, the guardband, and the usable acceptance zone.

Here is the same load cell, the same ±0.1 % tolerance on 10 000 lbf, the same measured value of 10 008 lbf, under two calibration sources. Against a secondary standard, U ≈ 4.03 lbf, k = 2, and the TUR is about 2.5:1. The curve spills past the upper limit, and the specific PFA at 10 008 lbf is about 16 %. To pull that below 2.5 % you need a guardband of about 3.95 lbf, which eats nearly 40 % of the tolerance and leaves an acceptance zone of ±6.05 lbf. Against deadweight primary standards, U ≈ 0.41 lbf, k = 2, the TUR is about 24:1, the guardband shrinks to about 0.40 lbf, and the acceptance zone is ±9.60 lbf, almost the full specification. The TUR did not make the decision. It set how much of the tolerance you get to keep.

TUR Is One of Two Doors, Not the Only One

There is a second route to the same 2 % global target, and it has nothing to do with TUR. NASA HDBK 8739.19-4 Annex 4 and the paper Using Reliability to Meet Z540.3's 2 % Rule show that if the end-of-period reliability for a family of like instruments is 89 % or better, the unconditional false accept risk across that population stays at or below 2 % whatever the measurement uncertainty. Our Start With the End in Mind paper calls these the dual dominance conditions: bench-level dominance at TUR ≥ 4.6:1, global dominance at EOPR ≥ 89 %. Either one alone is sufficient. That is why a lab with a modest TUR and good reliability records can meet a Z540.3 program requirement, and why a lab with a superb TUR and no records still has to compute the guardband. The two doors open onto the same room, and both are population statements. Neither changes the specific risk of the result at 10 008 lbf.

The Boundary

TUR is an input to the decision rule, and sometimes a precondition inside it. It is never the rule. Specifying TUR ≥ 4:1 on a purchase order with no decision rule tells the lab how capable to be and tells it nothing about how to treat a result at 10 008 lbf. The Decision Rule Guidance puts it this way: a stated TUR is a global risk control. Specific risk governs the correctness of an individual pass/fail decision. Both are legitimate. They answer different questions.

TUR ≥ 4:1 by itself is a wish. UKAS LAB 48 writes simple acceptance rules the same way: measurement uncertainty enters as a pre-condition that must be met before the acceptance decision, and its example rule reads PASS when the measured value is within tolerance and the capability constraint is met, FAIL otherwise.

-Henry Zumbrun, Morehouse Instrument Company

References

Sources cited across Parts 1 through 7, listed alphabetically. JCGM documents follow the citation form the BIPM recommends, with the member-organization author line and the JCGM DOI. Clause numbers given in the text refer to these editions.

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BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML. Evaluation of measurement data — The role of measurement uncertainty in conformity assessment. Joint Committee for Guides in Metrology, JCGM 106:2012. https://doi.org/10.59161/JCGM106-2012

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Morehouse Instrument Company. Start With the End in Mind. York, PA, 2026. https://mhforce.com/wp-content/uploads/2026/05/Start_With_The_End_in_Mind.pdf

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