TUR calculator: test uncertainty ratio, TAR and false-accept risk

The test uncertainty ratio (TUR) compares a tolerance with the expanded uncertainty of the calibration that checks it: TUR = tolerance span ÷ (2 × U). This calculator gives you the TUR and the TAR, and goes one step further: the probability that a calibration passes an instrument that is actually out of tolerance (false accept), with or without a guard band.

Don’t have U? Build it with our measurement uncertainty calculator: list the contributions, and it gives you the combined uncertainty, the coverage factor k and the expanded uncertainty U to enter here.

The calculations run in your browser and nothing is sent anywhere. They are checked by 58 automated tests against worked examples from Global ACI-TECH-1-002 (formerly ILAC G8), JCGM 106, NASA-HDBK-8739.19-4 and two NASA conference papers (assumptions and limits).

How to use the calculator

1. Tolerance. Enter the tolerance of the instrument you are calibrating, at the test point: as ± a value, as lower and upper limits, or as a manufacturer-style spec (% of reading + % of full scale). Leave one limit blank for a one-sided tolerance.
2. Uncertainty. Enter the expanded uncertainty U of the calibration and its coverage factor k, from the certificate or your uncertainty budget. If you don’t have U, tick the box and build a rough estimate from the reference standard, the resolution and the repeatability.
3. Optional: a measured value. The calculator tells you how likely it is that the true value is in tolerance, and whether the value is accepted under the rule you chose.
4. Optional: false-accept risk. Enter how often instruments of this type arrive in tolerance (the end-of-period reliability, EOPR) and choose an acceptance rule. You get the probability of false accept (PFA) and false reject (PFR) for that population.

What TUR is, and how to calculate it

Accreditation guidance defines the TUR as the tolerance limit divided by the 95 % expanded uncertainty of the measurement process (Global ACI-TECH-1-002, §2.13). The definition used in US practice since ANSI/NCSL Z540.3 is the same idea written for two-sided tolerances: the tolerance span divided by twice the 95 % expanded uncertainty (A2LA G136; NASA-HDBK-8739.19-4, §3.5.1). For a symmetric ±T tolerance, both give T / U.

Take the pressure gauge from our calibration certificate guide: tolerance ±0.75 psi, U = 0.12 psi (k = 2).

TUR = (2 × 0.75) ÷ (2 × 0.12) = 1.50 ÷ 0.24 = 6.25, reported as 6.2:1

We round the ratio down, never up, so it is never overstated. JCGM 106 expresses the same comparison as the measurement capability index Cm = tolerance span ÷ (4u). With k = 2 and a symmetric tolerance, Cm equals the TUR (JCGM 106, §7.6).

TUR vs TAR

The test accuracy ratio (TAR) is the older measure: the tolerance of the instrument under test divided by the accuracy specification of the reference standard. It ignores everything else that contributes to uncertainty. NASA’s decision-risk handbook uses TAR as another label for TUR, but in the traditional sense the two can be far apart (NASA-HDBK-8739.19-4, §1.4; Mimbs 2007).

A worked example from a NASA paper shows how far (Mimbs 2007, examples 1–3):

A 25:1 TAR looks generous. The TUR shows the calibration is closer to 3:1. The same paper warns that a TAR computed this way does not meet Z540.1 or Z540.3.

Where 4:1 comes from, and whether you need it

According to the NASA paper cited above, which draws partly on interviews with him, the 4:1 ratio goes back to work by J. Hayes and S. Crandon at the US Naval Ordnance Laboratory in the mid-1950s. Assuming a process spread of about ±2σ across the tolerance, they aimed at a consumer risk of about 1 %, got roughly 3:1, and padded it to 4:1 because they did not fully trust the stated tolerances of reference standards. It then spread through military purchasing standards (Mimbs 2007). JCGM 106 notes that ratio rules are sometimes ambiguously defined, and that the capability index Cm removes the ambiguity: Cm ≥ 4, for example, means u ≤ T/16 (JCGM 106, §7.6.4).

Where you will meet it today:

• ANSI/NCSL Z540.3 required a false-accept probability of no more than 2 % or, where that could not be estimated, a TUR of at least 4:1 (as summarized in NASA-HDBK-8739.19-4, §3.5, and Mimbs 2011). NCSL International withdrew Z540.3 in October 2020 and lists ISO/IEC 17025:2017 as its replacement. Contracts, especially military ones, may still invoke it (NCSLI; ANAB).
• ISO/IEC 17025 does not set a ratio. When a lab issues a statement of conformity, it needs a documented decision rule that takes uncertainty into account. A2LA describes a TUR of 4:1 or greater as the generally accepted industry standard, and gives simple acceptance with a TUR of 4:1 or greater as an example of an acceptable decision rule (A2LA G136).
• TUR below 1:1: the uncertainty is larger than the tolerance. A2LA notes that the risk of a wrong conformity call can then exceed 50 %, and that a lab should normally not accept such work (A2LA G136).

TUR isn’t risk: false accept depends on how often instruments are in tolerance

A TUR describes the calibration. The probability of a false accept also depends on the instruments being calibrated: if almost all of them arrive well within tolerance, few can be wrongly passed. That input is the end-of-period reliability (EOPR), the share of instruments found in tolerance at calibration. NASA’s handbook says that in many, if not most, cases the in-tolerance probability is the variable with the most influence on decision risk, and describes 4:1 as a crude risk-control tool (NASA-HDBK-8739.19-4, §3.5.2, §3.6).

What the chart shows:

• At TUR 4:1, the PFA stays at or below 2 % only when the EOPR is about 79 % or higher. That matches the value in Mimbs (2007, Fig. 3). It peaks at about 2.2 % for an EOPR near 65 %.
• At an EOPR of 95 %, the PFA is about 0.9 % at 4:1 and still only about 1.5 % at 1.5:1.
• The NASA paper’s vernier micrometer has a TUR of only 1.5:1, but an EOPR of 97.3 % keeps its PFA at about 0.9 % (Mimbs 2007). A low TUR is not automatically a high risk, and a high TUR is not automatically a low one.

Which EOPR is high enough? NASA’s current metrology standard accepts an observed EOPR as evidence of a false-accept probability of 2 % when there are at least 20 like calibrations (same model, procedure and interval): 95 % or higher for 20 to 39 calibrations, 93 % for 40 to 69 and 92 % for 70 or more (NASA-STD-8739.12A, 2024, §4.3, Table 1). An “89 % EOPR” figure still circulates; it comes from an earlier NASA policy described in 2011 (Mimbs 2011), not from the current standard.

What to do when TUR is below 4:1

1. Reduce the uncertainty. Use a better reference standard, more repeat readings, or better control of the environment. The calculator shows the largest U that still gives 4:1.

2. Use a guard band. Accept only results inside an acceptance limit that sits inside the tolerance. The table shows the maximum specific false-accept risk for a result sitting exactly on the acceptance limit (k = 2, normal distribution, only the nearer tolerance limit considered).

   Guard band w	Maximum specific PFA at the acceptance limit
   0 (simple acceptance)	50 %
   0.83 U	about 4.8 %
   U	about 2.3 %
   1.5 U	about 0.13 %
   3 U	less than 1 in a million

   Global ACI-TECH-1-002 lists these rules as upper bounds (§6.2, Table 1); the values above are the exact figures for k = 2. A guard band always increases false rejects (TECH-1-002, §6.4).

3. Quantify the risk instead of the ratio. If you know the EOPR from your calibration history, calculate the PFA directly (section 4 of the calculator). If you choose “guard band for a target PFA”, the calculator finds the smallest guard band that meets your target.

4. Look at the interval. A shorter calibration interval raises the EOPR, which lowers the PFA for the same TUR, in the usual range above about 70 % EOPR (NASA-HDBK-8739.19-4, §5.1).

5. Agree the decision rule up front. Under ISO/IEC 17025 (clause 7.1.3), the lab and the customer agree the rule before the work is done, unless it is built into the specification. See our guide to guard banding and decision rules.

Specific risk vs global risk

The calculator gives two kinds of risk, and they answer different questions (Global ACI-TECH-1-002, §2.14–2.15; JCGM 106, §9):

• Specific risk (section 3): for this one measured value, how likely is it that the true value is out of tolerance? It only needs the measured value and its uncertainty.
• Global risk (section 4): across all instruments calibrated this way, what fraction will be wrongly accepted or wrongly rejected? It also needs the EOPR.

A single instrument can pass a global criterion such as 2 % PFA and still have a specific risk near 50 %, if its result sits right on the limit (TECH-1-002, §6.3).

Assumptions and limits

• Normal distributions for the measurement error and for the population of instruments, the latter centered in the middle of the tolerance (the nominal value for a ± tolerance). This is the standard model in JCGM 106 (Annex A) and NASA-HDBK-8739.19-4.
• EOPR, true or observed. An observed in-tolerance rate already includes calibration uncertainty. Tick the “observed” box to correct it, following NASA’s method (NASA-HDBK-8739.19-4, eq. 4-17).
• Global risk needs a two-sided tolerance. For one-sided tolerances the calculator gives the specific risk only.
• Test value uncertainty is not the same as U. Some dimensional standards compute their capability index with a “test value uncertainty” that leaves out contributions from the instrument under test. A risk computed from it does not describe the real risk of a wrong conformity statement (DKD-L 4-1).
• The uncertainty budget option is a rough estimate. It adds standard uncertainties in quadrature, with rectangular distributions for specifications (a/√3) and resolution (δx/√12) (GUM 4.3.7, F.2.2.1). A real budget usually has more terms.
• Validation. The math is checked against worked examples from Global ACI-TECH-1-002, formerly ILAC G8 (Table 1), JCGM 106 (conformance probability, measurement capability index and global risk examples), NASA-HDBK-8739.19-4 (Tables 5-1, 5-2 and 5-4, Appendix B) and the NASA papers cited. All 58 automated checks pass: worked examples agree with the published values to the precision the sources give, and bound-type values (Table 1) fall within the stated bounds.