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Technical article · Metrology

Proof of ± 0.1 K: measurement uncertainty, traceability and acceptance

An accuracy requirement in the customer specification becomes an agreement that you can examine only with a decision rule. Without a decision rule, the risk of false acceptance at the limit is up to 50 %.

Context

Accuracy requirement in the customer specification

“Flow temperature ± 0.1 K” is a typical accuracy requirement in customer specifications for test benches. The requirement leaves 3 questions open:

  • Does the requirement apply to the control deviation or to the measurement uncertainty?
  • Does it apply to the setpoint or to the indication?
  • At the acceptance, how do you show that the test bench meets the requirement? The measurement chain itself has an uncertainty of the same magnitude.

This article gets the answers from the documents that exist for this purpose. All of them are freely available. The article explains the terms and shows the uncertainties that are possible for each measured quantity. It describes the decision rule. Without this rule, an acceptance gives no result.

01

Terms in JCGM 100:2008

The decisive document is the Guide to the expression of uncertainty in measurement (GUM), JCGM 100:2008. The publisher is the Joint Committee for Guides in Metrology (JCGM). The International Bureau of Weights and Measures (BIPM) gives the guide free of charge[1] [2]. The guide defines the measurement uncertainty as a “parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand”[2].

Terms in JCGM 100:2008
TermDefinition
standard uncertaintyuncertainty, expressed as a standard deviation[2]
combined standard uncertaintypositive square root of the weighted sum of the variances of the input quantities[2]
expanded uncertainty Uinterval about the measurement result that contains a large fraction of the distribution of values[2]
coverage factor kfactor that gives U from the combined standard uncertainty, typically between 2 and 3[2]
Type A evaluationby statistical analysis of measured values[3]
Type B evaluationby other means, for example from calibration certificates, drift data or accuracy classes[4]

Type A and Type B are different in the method of evaluation, not in the type of component. For test bench construction, this means: repeated measurements on the installed plant are Type A. Manufacturer specification, class, calibration certificate and resolution are Type B.

For input quantities without correlation, the law of propagation of uncertainty applies. The combined variance is the sum of the squared contributions. Each contribution has a weight, its sensitivity coefficient. This coefficient is the partial derivative of the model with respect to the input quantity[2]. For quantities with correlation, covariance terms add to the sum.

uc²(y) = Σ ci² · u²(xi) with ci = ∂f / ∂xi

For the expanded uncertainty, this equation applies:

U = k · uc

The standard value k = 2 gives a coverage of approximately 95 %[2] [13].

JCGM GUM-1:2023 and JCGM 100:2008

JCGM GUM-1:2023 does not replace the GUM. It replaces only the earlier introduction JCGM 104:2009[5]. JCGM 100:2008 stays valid and has the name GUM-3 in the new nomenclature[6]. An amendment to JCGM 100:2008 for non-linear models is in work[1]. For strongly non-linear models, the Monte Carlo method in JCGM 101 is the more robust tool[7]. An example is the flow through an orifice plate with its square-root relation.

02

Traceability and calibration certificate

The International Vocabulary of Metrology (VIM) defines metrological traceability as the “property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty”[8]. 2 notes from this entry are important for test benches:

  • For measurements with more than one input quantity, each input quantity must be traceable, not “the test bench” as a whole.
  • Traceability alone does not guarantee a suitable measurement uncertainty or the absence of mistakes[8].

Calibration and adjustment

Calibration is not adjustment. The vocabulary makes this clear: do not confuse calibration with the adjustment of a measuring system. People often call the adjustment “self-calibration”, and this name is wrong[9].

The ILAC policy on metrological traceability specifies the permitted routes to traceability[10]. ILAC is the International Laboratory Accreditation Cooperation. The policy prefers 2 routes:

  • calibration by a national metrology institute whose service is in the scope of the mutual recognition arrangement of the Metre Convention
  • calibration by a calibration laboratory whose service is in the scope of the ILAC arrangement

Other routes apply only if these 2 routes are not possible[10]. One sentence of the ILAC policy has a direct effect on the calibration certificate:

Reference to accreditation on the certificate

Only certificates with the accreditation symbol or a text reference to the accreditation of the calibration laboratory get the full recognition[10].

The German accreditation body (DAkkS) does not accept result reports of accredited calibration laboratories without a reference to the accreditation as evidence of traceability[11]. The same rule specifies that the measurement ranges and uncertainties in the scope of accreditation are not flexible. The scope applies exactly to the given points.

The ILAC policy on measurement uncertainty has a requirement for the calibration certificate. The measurement result contains the value of the measured quantity and the related expanded uncertainty. This value applies to a coverage probability of approximately 95 %. Accredited laboratories must not give measurement uncertainties smaller than their own calibration and measurement capability[12].

2 checks on the calibration certificate

The 2 checks take approximately 30 s.

  1. Does the calibration certificate give a measurement uncertainty? If not, the certificate is not usable as evidence of traceability.
  2. Does it have the accreditation symbol or a text reference?

A factory calibration certificate is a warning sign if it gives a smaller uncertainty than an accredited calibration laboratory offers for the same quantity.

ISO/IEC 17025:2017 gives the requirements for the laboratory itself[14]. Clause 6.5 is about metrological traceability, clause 7.6 about the evaluation of measurement uncertainty. Clause 7.7 is about the assurance of validity through intermediate checks. Clauses 7.8.6.1 and 7.8.6.2 are about statements of conformity and the decision rule[10] [11].

03

Decision rule in accordance with JCGM 106 and ILAC-G8

The proof of ± 0.1 K does not depend on one measured value but on an agreed decision rule. JCGM 106:2012 defines the terms for this[15]:

Terms of conformity assessment in JCGM 106:2012
TermDefinition
tolerance limitspecified upper or lower limit of permitted values of a property
acceptance limitspecified upper or lower limit of permitted measured values
guard bandinterval between a tolerance limit and the related acceptance limit
decision ruledocumented rule that tells how to consider the measurement uncertainty for acceptance or rejection
consumer's riskprobability that the inspection accepts a non-conforming item
producer's riskprobability that the inspection rejects a conforming item
TOLERANCE LIMIT TL ACCEPTANCE LIMIT AL PERMITTED RANGE NOT PERMITTED Guard band w = r · U accepted in the tolerance, but in the guard band: rejected rejected MEASURED VALUE WITH EXPANDED UNCERTAINTY U (k = 2) With simple acceptance, w = 0 and the acceptance limit is on the tolerance limit. At the limit, the risk of false acceptance is up to 50 per cent. With w = U, the risk decreases below 2.5 per cent, with w = 0.83 · U below 5 per cent.
Fig. 1 · Decision rule. The guard band w moves the acceptance limit inwards from the tolerance limit. The rule accepts a measured value only if it is within the acceptance limit. With w = 0, the 2 limits are the same: shared risk. Terms from JCGM 106:2012[15], numerical values from ILAC-G8:09/2019[16].

The guideline ILAC-G8 lists the usual rules with the risk of false acceptance[16]:

Decision rules and risk of false acceptance in ILAC-G8:09/2019
RuleGuard bandRisk of false acceptance
shared riskw = 0up to 50 % at the limit
ILAC-G8 rulew = Ubelow 2.5 %
in accordance with ISO 14253-1w = 0.83 · Ubelow 5 %
“six sigma”w = 3 · Ubelow 1 ppm

Acceptance band at U = 0.05 K

With shared risk, a measured value within ± 0.1 K is sufficient. Exactly at the limit, a residual risk of up to 50 % remains. If the customer requires a smaller risk, the proof applies against 0.1 K minus U. Then the risk of false acceptance is below 2.5 %. With a measurement chain with U = 0.05 K, an acceptance band of ± 0.05 K remains.

This calculation belongs before the contract, not into the acceptance. ISO/IEC 17025 explicitly requires a clear definition of the specification and the decision rule and an agreement with the customer[16].

For the inspection of measuring equipment and workpieces, ISO 14253-1:2017 applies. With the edition of 2013, the title changed from “proving” to “verifying”. The current edition speaks of the verification of conformity, not of proof[17] [18].

In the automotive supply chain, the capability of test processes follows the same system. There, the capability ratio comes from the expanded uncertainty of the measuring system and the tolerance. The limit is 15 %. The guard distance is g = 1.65 · u. This agrees with a risk of false acceptance of 5 % on each side and thus agrees with ISO 14253-1[19]. The related international standard is ISO 22514-7:2021[20].

04

Possible uncertainty for each measured quantity

Temperature

For platinum resistance thermometers, IEC 60751 applies in the 3rd edition of 2022[21]. The tolerance classes are equations that depend on the temperature, not fixed values:

Tolerance classes for platinum resistors in IEC 60751:2022
ClassToleranceat 0 °Cat 100 °C
W 0.1 / F 0.1± (0.1 + 0.0017 · |t|) °C± 0.10 K± 0.27 K
W 0.15 / F 0.15± (0.15 + 0.002 · |t|) °C± 0.15 K± 0.35 K
W 0.3 / F 0.3± (0.3 + 0.005 · |t|) °C± 0.30 K± 0.80 K
W 0.6 / F 0.6± (0.6 + 0.01 · |t|) °C± 0.60 K± 1.60 K

Class name and need for calibration

  1. The names “1/3 DIN” and “1/10 DIN” are not names from the standard. IEC 60751 does not contain them[22].
  2. Also class W 0.1 / F 0.1 gives only ± 0.27 K at 100 °C. For ± 0.1 K, a calibration of the installed sensors is thus necessary. The class alone is not sufficient.

For the uncertainty budget, there is the calibration guideline DKD-R 5-1, edition 2023, of the German Calibration Service (DKD). It is freely available. It gives specific numbers for contributions that you otherwise must estimate[23]:

Uncertainty contributions for resistance thermometers in DKD-R 5-1
ContributionValue if not measured
self-heatingrectangular distribution with half-width 30 mK, thus u ≈ 17 mK
hysteresis0.2 % of the full measurement range of the thermometer in kelvin, in many cases the largest contribution
long-term drifttypically below 3 mK per year, as rectangular distribution
parasitic thermoelectric voltage25 µV at 1 mA measurement current agree with approximately 63 mK
insulation resistance1 MΩ at 400 °C gives a contribution of 0.177 K
immersion depthat least 10 times the sensor diameter

Part of the budget for the sensor

If you did not measure the hysteresis, the general value of 0.2 % of the measurement range applies[23]. For a measurement range of 100 K, this is 0.2 K. This is more than the full requirement of ± 0.1 K. Thus a commitment to ± 0.1 K needs a measured hysteresis. Also then, the sensor uses a part of the budget before you connect the input card. The contributions add in quadrature[2].

For thermocouples, IEC 60584-1 in the edition of 2013 applies[24]. The tolerances are class values[25]:

  • Type T, class 1: the larger value of ± 0.5 °C and 0.004 · |t|
  • Type T, class 2: the larger value of ± 1.0 °C and 0.0075 · |t|
  • Type K, class 1: the larger value of ± 1.5 °C and 0.004 · |t|

The same source documents effects that occur in daily test bench operation[25]:

  • green rot of type K in an atmosphere with low oxygen between approximately 800 and 1 050 °C
  • a reversible error of the thermoelectric voltage of type K if the sensor cools faster than approximately 100 K per hour between 600 and 400 °C
  • the errors of thermocouple and compensation cable add

Pressure

For Bourdon tube pressure gauges, EN 837-1 applies[26]. The decisive property is the reference of the class. The class refers to the span, not to the measured value[27]. A class 1.6 gauge with a range of 0 to 100 bar has ± 1.6 bar over the full range. At the operating point 10 bar, this is 16 % of the measured value.

For the calibration, the guideline DKD-R 6-1 is decisive. It is freely available[28]. It defines 3 calibration sequences for different target uncertainties:

As contributions, the guideline gives resolution, uncertainty of the standard, zero deviation, repeatability, hysteresis and reproducibility[28]. The pressure balance gives the smallest possible uncertainty. It is possible to find its effective piston area with relative uncertainties below 5 · 10⁻⁶, and the applied force with 1 · 10⁻⁶ or better[29].

Flow

For differential pressure devices, the series ISO 5167, parts 1 to 4 in the edition 2022, applies[30] [31]. A value from the previous edition of part 2 shows the magnitude of the possible uncertainty. For diameter ratios between 0.20 and 0.60, the relative uncertainty of the discharge coefficient alone is 0.5 %. Additional terms for small pipe diameters and low Reynolds numbers add arithmetically[32].

Limit of the standard orifice plate

From the discharge coefficient alone, a standard orifice plate gives at least 0.5 % uncertainty. This applies only if the installation meets all geometric conditions and inlet conditions exactly. For requirements in the range of tenths of a per cent, it is not suitable.

For Coriolis meters, there is the standard ISO 10790 for selection and installation[33]. In addition, there is a freely available guide of a national test laboratory with reliable numbers from practice[34]. Manufacturer data start at 0.05 % for the mass flow. The guide gives the conditions for this[34]:

  • The zero stability has a large proportional effect at small flows. Thus an oversized meter at part load increases the uncertainty.
  • Temperature makes the measuring tubes softer and causes an indication that is too high. Pressure increases the stiffness and causes an indication that is too low. The 2 effects are linear.
  • A calibration at a viscosity of 50 mm²/s causes deviations above 0.5 % in an application at 300 mm²/s.

The main recommendation of the guide is: calibrate as near to the operating conditions as possible. The Physikalisch-Technische Bundesanstalt (PTB) gives a target uncertainty of ± 0.02 % for its hydrodynamic test field[35]. This value shows the smallest uncertainty that the national metrology institute aims for in this test field.

Electrical power

For electrical power, there are no accuracy classes as for pressure gauges. The manufacturer specification is decisive. A precision power analyser is an example at the top end[36]:

Limits of this specification

  1. At part load, the part “of range” is dominant. At 10 % of the range, its part of the measured value is 10 times larger.
  2. The data apply at power factor 1. For converter loads with a small power factor, the values are much worse.
  3. The data apply without a current transformer. In a real test bench, the transformer is usually the dominant contribution.
05

Uncertainty budget of a heat output

A derived characteristic value in test bench construction is the heat output from mass flow, specific heat capacity and temperature difference:

Q̇ = ṁ · cp · ΔT

For a pure product, the relative uncertainties add in quadrature. The sensitivity coefficients cancel out[2].

(u(Q̇) / Q̇)² = (u(ṁ) / ṁ)² + (u(cp) / cp)² + (u(ΔT) / ΔT)²
0 2 4 6 8 10 12 5 10 15 20 25 30 TEMPERATURE SPREAD ΔT · K REL. UNCERTAINTY OF Q · % u(ΔT) = 0.20 K u(ΔT) = 0.10 K u(ΔT) = 0.05 K Contribution of mass flow and fluid property alone: 0.64 % Model Q = ṁ · c · ΔT with u(ṁ)/ṁ = 0.4 % and u(c)/c = 0.5 %, k = 1
Fig. 2 · Relative uncertainty of the heat output as a function of the temperature spread, for 3 uncertainties of the temperature difference. Below a spread of approximately 10 K, the temperature term is larger than all other terms. Our own calculation with the law of propagation of uncertainty[2], with an assumed 0.4 % for the mass flow and 0.5 % for the fluid property.

The curve shows: the uncertainty of the temperature difference is an absolute quantity, and the spread is in the denominator. At a spread of 5 K and an uncertainty of 0.1 K, this gives 2 % on the output. This is more than mass flow and fluid property together. At 30 K, it is 0.33 %.

Spread before flow measurement

If you want to decrease the uncertainty of a heat output measurement, first increase the spread, not the effort for the flow measurement.

Correlation of sensor pairs

Flow and return sensors with a common calibration against the same standard and through the same input card have correlated deviations. The covariance term of the propagation law then decreases the uncertainty of the difference by the factor √2, compared with the calculation without correlation[2]. For this reason, legal metrology certifies sensor pairs and not single sensors.

The technical guideline for the verification of heat meters shows the system behind this[37]. It divides the meter into the 3 terms of the model: flow sensor, calculator and temperature sensor pair. For each term it gives a maximum permissible error and adds these errors arithmetically. Maximum permissible errors are not uncertainties. The 2 terms that depend on the temperature contain the ratio of the smallest spread to the measured spread. Thus they show the dominance from Fig. 2[37].

The same guideline documents a rule that plant construction usually gives on only by word of mouth. The measurement uncertainty of the test equipment must not be more than a specified fraction of the maximum permissible error at the test points. In accordance with the design, this is one fifth or one third[37].

The document EA-4/02 of European Accreditation (EA) contains fully calculated budgets as examples and is freely available[13]. The examples include the calibration of a thermocouple at 1 000 °C, of a temperature block calibrator at 180 °C and of a water meter.

For calorimetric measurements, there is a freely accessible report of the National Institute of Standards and Technology (NIST). It covers a 20 MW calorimeter and fully calculates 2 models with tabulated contributions in per cent[38].

06

Calibration intervals and intermediate checks

The decisive guideline is ILAC-G24, identical to OIML D 10, in the edition 2022. OIML is the International Organization of Legal Metrology. The guideline is freely available[39]. Since this edition, its title is “recalibration intervals of measuring equipment”. The older title of 2007 is out of date.

There are 5 methods[39]:

  1. automatic adjustment by calendar time
  2. control chart
  3. interval in operating hours instead of calendar months
  4. frequent check of critical parameters, full calibration only when a parameter is outside its limit
  5. other statistical methods

The guideline states explicitly: each laboratory decides itself which method it applies. The basis is its needs and its risk assessment[39].

Reference measurement during operation

For a test bench, the 4th method is the obvious selection. A test bench with a defined reference point can do a reference measurement during operation. For laboratories in accordance with ISO/IEC 17025, intermediate checks are part of the standard when they are necessary[14].

For this, the test bench needs a defined reference point in the circuit, a stored setpoint and an evaluation that records the trend over time. With this equipment, you can extend the calibration intervals with evidence. A drift becomes visible before it makes a test result incorrect.

07

Content of an acceptance agreement

From sections 01 to 06 comes a checklist. It is a conclusion of Profectronics.

  1. Separate the quantity that you mean. Control deviation, measurement uncertainty and reproducibility are 3 different numbers. A requirement “± 0.1 K” without the quantity that it applies to is not verifiable.
  2. Agree on the decision rule. Shared risk or guard band. For a guard band, specify the factor[15] [16].
  3. Name the measurement chain. Sensor, input card, calibration laboratory and scope of calibration. Traceability applies to each input quantity, not to the test bench as a whole[8].
  4. Attach the uncertainty budget. Use the example of EA-4/02: quantity, estimate, standard uncertainty, distribution, sensitivity coefficient, contribution[13].
  5. Specify test points and conditions. Temperature, flow and ambient condition for the proof.
  6. Define intermediate checks. Item, frequency, reference and action when a value is outside its limit[39].
  7. Examine calibration certificates for measurement uncertainty and reference to accreditation[10] [11] [12].

For acceptance measurements on plants with redundant measurement points, there is one more tool. This tool is data reconciliation with balance equations in accordance with VDI 2048 Part 1[40]. If energy balances and mass balances close, it can decrease the uncertainty, in some cases by a large amount. It also finds incorrect measurement points. Data reconciliation needs redundant measurement points and closed energy balances and mass balances.

Research status August 2026

Numerical values from standards that are not free of charge come from preview versions and from manufacturer documents that quote these standards. This applies to IEC 60751, IEC 60584-1, EN 837-1 and ISO 5167. Profectronics builds test benches and is not a calibration laboratory.

Before you use these values as a commitment, compare them with the purchased copy of the standard.

08

Sources

The basic documents of metrology are available free of charge: GUM, VIM, EA-4/02, the DKD guidelines, ILAC-G8 and OIML D 10.

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Decision rule in the customer specification

If you write a customer specification, specify the decision rule before you write the accuracy requirement into it.

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