Uncertainty
Uncertainty budget templates
A starting list of the contributions each calibration usually has, with the distribution the GUM assigns to that kind of knowledge and why. Open a template in the budget calculator, replace the numbers with your own evidence, and check the result with the Monte Carlo method.
Illustrative values, not a budget for your laboratory. Replace every number with your own evidence, remove rows that do not apply and add the ones your method has. The result is a calculated value, not a certified result.
Temperature
Pt100 by comparison in a liquid bath
A platinum resistance thermometer (or any probe with an indicator) is compared with a reference PRT in a stirred liquid bath.
Measurand: Correction to the UUT temperature at one point, in °C.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the comparison | A · s/√n (ν = 5) | 6 readings, u = 0.000601 °C | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated differences UUT − reference at the point. |
| Reference PRT certificate | B · Normal, stated U and k | U = 0.01, k = 2 °C | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | U and k at this temperature from the PRT’s certificate. |
| Reference PRT drift since calibration | B · Rectangular (uniform) | ± 0.005 °C | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | The largest change between successive calibrations, or the maker’s stability figure. |
| Bath stability during the readings | B · U-shaped (arcsine) | ± 0.005 °C | A controlled medium cycles between limits around its set point; the arc sine (U-shaped) distribution fits a value that spends most time near the extremes (JCGM 101 6.4.6). | Half the peak-to-peak variation of the reference over the reading period. |
| Bath uniformity (reference to UUT position) | B · Rectangular (uniform) | ± 0.004 °C | Difference between where the reference and the UUT sit, bounded by the surveyed gradient, rectangular (GUM 4.3.7). | Half the largest gradient across the working volume from your uniformity survey. |
| Reference indicator (bridge) certificate | B · Normal, stated U and k | U = 0.003, k = 2 °C | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | The bridge or readout certificate, converted to temperature at the PRT’s sensitivity (about 0.39 Ω/°C for Pt100). |
| UUT indicator resolution | B · Rectangular (uniform) | ± 0.005 °C | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. | Half the UUT display step (0.005 °C for a 0.01 °C display). |
| Self-heating difference | B · Rectangular (uniform) | ± 0.002 °C | The measuring current warms both sensors; what matters is the difference between calibration and use conditions, bounded and rectangular. | From a two-current test, or the maker’s self-heating figure at your current. |
With these illustrative values the GUM budget gives u_c = 0.00797 °C, U = 0.0159 °C (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- JCGM 101:2008 6.4.6: arc sine distribution for a cycling controlled temperature.
- Structure of a comparison calibration in a liquid medium as described in the free EURAMET calibration guides for temperature (euramet.org); no table or example reproduced.
- Caveat: Hysteresis of the UUT and stem-conduction (immersion) errors are not listed; add rows if your method does not eliminate them.
- Caveat: If the UUT is a bare sensor read on your own bridge, replace the UUT resolution row with your bridge’s certificate and resolution.
Temperature
Thermocouple by comparison in a furnace or block
A base-metal thermocouple is compared with a reference thermometer in a furnace, dry block or bath, its EMF read on a voltmeter with a known reference junction.
Measurand: Correction to the UUT temperature at one point, in °C.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the comparison | A · s/√n (ν = 4) | 5 readings, u = 0.0121 °C | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated differences at the point. |
| Reference thermometer certificate | B · Normal, stated U and k | U = 0.1, k = 2 °C | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | From the reference’s certificate at this temperature. |
| Reference drift since calibration | B · Rectangular (uniform) | ± 0.05 °C | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | Your history for the reference. |
| Reference junction temperature | B · Rectangular (uniform) | ± 0.05 °C | An ice point or an electronic compensation holds the reference junction within a bound; the error passes to the hot-junction temperature almost one to one for a base-metal couple (use the thermocouple tool to convert if needed). | The ice-point or compensator specification. |
| Furnace or block uniformity | B · Rectangular (uniform) | ± 0.15 °C | Difference between where the reference and the UUT sit, bounded by the surveyed gradient, rectangular (GUM 4.3.7). | From your uniformity survey at this temperature. |
| Furnace or block stability | B · U-shaped (arcsine) | ± 0.05 °C | A controlled medium cycles between limits around its set point; the arc sine (U-shaped) distribution fits a value that spends most time near the extremes (JCGM 101 6.4.6). | Half the peak-to-peak variation over the reading period. |
| Voltmeter certificate and resolution, as temperature | B · Normal, stated U and k | U = 0.05, k = 2 °C | The voltmeter’s U at the EMF read, divided by the thermocouple’s Seebeck coefficient at the point (about 41 µV/°C for type K near 300 °C), normal. | Voltmeter certificate at the EMF read, converted with the thermocouple tool. |
| UUT wire inhomogeneity | B · Rectangular (uniform) | ± 0.2 °C | A thermocouple’s EMF depends on the temperature gradient along inhomogeneous wire; a bound from an immersion test, rectangular. | Change in reading when the immersion is varied, or a conservative figure for the wire type. |
With these illustrative values the GUM budget gives u_c = 0.164 °C, U = 0.329 °C (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- Structure of thermocouple comparison calibration as described in the free EURAMET calibration guide for thermocouples (euramet.org); no table reproduced.
- Seebeck coefficients from the NIST ITS-90 reference functions (the site’s thermocouple tool).
- Caveat: Inhomogeneity usually dominates for used base-metal couples; measure it rather than assume it.
- Caveat: For noble-metal couples the reference-junction and voltmeter rows get larger in temperature terms because the Seebeck coefficient is smaller.
Electrical
Digital multimeter, DC voltage against a calibrator
A multifunction calibrator sources a DC voltage and the DMM’s reading is compared with it.
Measurand: Error of the DMM reading at one point, in V.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the DMM readings | A · s/√n (ν = 4) | 5 readings, u = 0.00000374 V | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated readings at the point. |
| Calibrator certificate | B · Normal, stated U and k | U = 0.00003, k = 2 V | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | The calibrator’s certificate at this range and value. |
| Calibrator one-year specification (drift) | B · Rectangular (uniform) | ± 0.00005 V | If you rely on the maker’s specification rather than the certificate, its ± limit is a rectangular bound (GUM 4.3.7). Use one or the other for the same effect, not both. | The accuracy specification for the period since calibration, as an absolute value at this point (the spec-to-tolerance tool converts % of reading + floor). |
| DMM resolution | B · Rectangular (uniform) | ± 0.000005 V | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. | Half the last digit on the range used. |
| Thermal EMFs in the leads | B · Rectangular (uniform) | ± 0.000002 V | Dissimilar-metal junctions at the terminals add a small offset; bounded, rectangular. Reversing the leads and averaging removes most of it. | The offset seen with the calibrator at zero, or a conservative bound for your leads. |
| Loading and lead effects | B · Rectangular (uniform) | ± 0.000001 V | The DMM input resistance loads the calibrator output; the error is V·R_source/R_in, bounded, rectangular. | Calculated from the calibrator output resistance and the DMM input resistance. |
With these illustrative values the GUM budget gives u_c = 0.0000329 V, U = 0.0000658 V (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- EA-4/02 M:2022 (EURAMET, free) contains a worked digital multimeter example of comparable structure; it is not reproduced here.
- Caveat: Use the calibrator certificate or its specification for the calibrator’s own error, not both.
- Caveat: At high voltage or on AC, add rows for the divider or the AC–DC difference.
Torque & Force
Hand torque wrench on a reference transducer
A setting-type or indicating hand torque tool is loaded against a reference torque transducer and indicator.
Measurand: Deviation of the wrench setting from the applied torque at one point, in N·m.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the wrench | A · s/√n (ν = 4) | 5 readings, u = 0.121 N·m | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated releases or readings at the point (ASME B107.300 and ISO 6789 set the number of repeats; use theirs). |
| Reference transducer and indicator certificate | B · Normal, stated U and k | U = 0.1, k = 2 N·m | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | From the transducer system’s certificate at this torque. |
| Reference drift since calibration | B · Rectangular (uniform) | ± 0.05 N·m | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | Your history, or the maker’s long-term stability figure. |
| Indicator resolution | B · Rectangular (uniform) | ± 0.05 N·m | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. | Half the indicator’s display step at this range. |
| Reproducibility (re-mounting, operator) | B · Rectangular (uniform) | ± 0.2 N·m | Changing the square-drive engagement, the loading point on the handle or the operator shifts the result; bounded by observation, rectangular. | Spread seen when the wrench is remounted and reloaded, from your method validation. |
| Temperature effect on the transducer | B · Rectangular (uniform) | ± 0.02 N·m | The transducer’s temperature coefficient times the departure from its calibration temperature, rectangular. | Coefficient from the transducer data sheet times your laboratory’s temperature range. |
| Loading point and side-load | B · Rectangular (uniform) | ± 0.1 N·m | Force applied off the marked point or off-axis changes the torque the wrench sees; bounded by the fixture’s geometry, rectangular. | From your fixture’s alignment tolerance and handle marking. |
With these illustrative values the GUM budget gives u_c = 0.189 N·m, U = 0.398 N·m (k = 2.11, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- Test method and number of repeats: ASME B107.300 and ISO 6789 (purchase from the publishers; not reproduced).
- Structure of torque measuring device uncertainty as described in the free EURAMET calibration guide for torque (euramet.org); no table reproduced.
- Caveat: For a setting-type wrench the measurand is the release torque; the repeatability row then holds the released values.
- Caveat: Clockwise and counter-clockwise are separate budgets if the wrench is calibrated both ways.
Dimensional
Gauge block by comparison (GUM H.1 structure)
An end gauge is compared with a calibrated standard of the same nominal length on a comparator, both at about 20 °C.
Measurand: Deviation of the gauge block’s length from nominal at 20 °C, in nm.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the measured difference | A · s/√n (ν = 4) | 5 readings, u = 1.36 nm | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated comparator differences (standard − UUT). |
| Calibration of the standard | B · Normal, stated U and k | U = 45, k = 2 nm | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | U and k from the standard’s certificate for this length. |
| Comparator random effects (from history) | B · Normal, stated U and k | U = 6, k = 1, ν = 24 nm | The comparator’s random error established from a pooled set of earlier measurements, used as a standard uncertainty with its degrees of freedom (GUM H.1.3.2). | Pooled standard deviation of your comparator checks and its degrees of freedom. |
| Comparator systematic effects | B · Normal, stated U and k | U = 4, k = 1, ν = 8 nm | Residual systematic error of the comparator after correction, from its own calibration (GUM H.1.3.2). | From the comparator’s calibration. |
| Thermal expansion coefficient of the standard | B · Rectangular (uniform) | ± 1.2 nm | The uncertainty of α_s (about ±2 × 10⁻⁶ /K, rectangular) times the length and the departure from 20 °C; here converted to nm already (GUM H.1.3.3). | Your α uncertainty × nominal length × |t − 20 °C| (the thermal expansion tool gives the row). |
| Temperature of the comparator bed | B · Normal, stated U and k | U = 3.5, k = 1 nm | Mean departure of the bed from 20 °C with its uncertainty, times L·α (GUM H.1.3.4). | From your thermometer’s certificate and the bed survey, times L·α. |
| Cyclic variation of the bed temperature | B · U-shaped (arcsine) | ± 5 nm | The thermostatically controlled bed cycles about its mean; an arc sine distribution for the cyclic term (GUM H.1.3.4), times L·α. | Half the amplitude of the cycle, times L·α. |
| Difference in expansion coefficients | B · Rectangular (uniform) | ± 2.9 nm | δα between standard and UUT, bounded, rectangular, times L·(t − 20) (GUM H.1.3.5). | Your bound on δα × L × |t − 20 °C|. |
| Difference in temperature between blocks | B · Rectangular (uniform) | ± 8 nm | δθ between the two blocks, bounded, rectangular, times L·α (GUM H.1.3.6). | Your bound on δθ × L × α. |
With these illustrative values the GUM budget gives u_c = 24.7 nm, U = 49.4 nm (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM) Annex H.1, Example: end-gauge calibration. The structure and the choice of distributions follow H.1.3; the illustrative numbers here are placeholders in nanometres, not the GUM’s figures.
- Caveat: The GUM example also carries second-order terms (H.1.7); add them when δα and δθ are both large.
- Caveat: Rows are given in nm after multiplying by the sensitivity coefficients; enter your own products or use the thermal expansion tool, which outputs rows with sensitivities.
Mass & Balances
Weight by substitution against a reference weight
A test weight is compared with a reference weight of the same nominal value on a mass comparator by ABBA or ABA substitution.
Measurand: Conventional mass correction of the test weight, in mg.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the weighing differences | A · s/√n (ν = 5) | 6 readings, u = 0.000601 mg | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your ABBA (or ABA) differences at this nominal value. |
| Reference weight certificate | B · Normal, stated U and k | U = 0.03, k = 2 mg | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | U and k of the reference’s conventional mass from its certificate. |
| Reference weight instability | B · Rectangular (uniform) | ± 0.01 mg | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | Change between successive calibrations of the reference. |
| Balance resolution | B · Rectangular (uniform) | ± 0.005 mg | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. In a substitution it enters twice (two readings per difference), so the half-width is √2 times half a digit. | √2 × half the comparator’s scale interval d. |
| Balance eccentricity | B · Rectangular (uniform) | ± 0.004 mg | Off-centre loading changes the indication; bounded by the eccentricity test, rectangular. | From your eccentricity test at this load. |
| Balance sensitivity | B · Rectangular (uniform) | ± 0.002 mg | The indicated difference is scaled by the balance sensitivity, whose uncertainty applies to the measured difference, rectangular. | Sensitivity uncertainty × the difference measured. |
| Air buoyancy correction | B · Rectangular (uniform) | ± 0.008 mg | The correction (ρ_a − ρ₀)·(V_t − V_r) is uncertain through the air density and the volumes; bounded, rectangular. The air density tool gives ρ_a and the buoyancy tool the correction. | Propagate u(ρ_a) and u(V) through the correction; small when both weights are the same alloy. |
With these illustrative values the GUM budget gives u_c = 0.0172 mg, U = 0.0344 mg (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- OIML R 111-1:2004 Annex C describes the uncertainty calculation for weight calibration (weighing process, reference weight, air buoyancy, balance); the structure is followed, nothing reproduced.
- Caveat: For weights of different material (stainless steel against brass or cast iron) the buoyancy row can dominate; measure air density rather than assume it.
- Caveat: Magnetic effects are not listed; add a row when the weights or the comparator pan are susceptible.
Pressure
Pressure gauge against a reference (deadweight tester or transducer)
A dial or digital pressure gauge is compared with a deadweight tester or a reference transducer on a common manifold, rising and falling.
Measurand: Error of the gauge reading at one point, in kPa.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the gauge readings | A · s/√n (ν = 4) | 5 readings, u = 0.051 kPa | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated readings at the point. |
| Reference certificate | B · Normal, stated U and k | U = 0.15, k = 2 kPa | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). | From the deadweight tester or transducer certificate at this pressure (the deadweight tester tool gives the pressure from the masses). |
| Reference drift since calibration | B · Rectangular (uniform) | ± 0.05 kPa | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | Your history for the reference. |
| Height (head) correction | B · Rectangular (uniform) | ± 0.03 kPa | The reference and the UUT sense pressure at different heights; the head correction (ρ·g·Δh) is uncertain through Δh and the fluid density, rectangular (the liquid head tool computes it). | u(Δh) × ρ × g, plus the density uncertainty for liquid media. |
| UUT resolution | B · Rectangular (uniform) | ± 0.25 kPa | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. For a dial gauge, half the smallest readable fraction of a division. | Half the display step, or half of what you can read between dial graduations. |
| UUT hysteresis | B · Rectangular (uniform) | ± 0.3 kPa | Difference between rising and falling readings at the same point; half of it as a rectangular half-width (GUM F.2.2.2). | Half the rising–falling difference at the point. |
| Temperature of the reference | B · Rectangular (uniform) | ± 0.02 kPa | The reference’s temperature coefficient (for a piston gauge, the thermal expansion of the piston–cylinder area) times the departure from its reference temperature, rectangular. | Coefficient × your temperature range at the reference. |
With these illustrative values the GUM budget gives u_c = 0.246 kPa, U = 0.492 kPa (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- Structure of pressure gauge calibration as described in the free EURAMET calibration guide for pressure gauges (euramet.org) and, for the reference, EURAMET cg-3 for pressure balances; no table reproduced.
- Caveat: If the UUT is read at rising pressure only and reported so, drop the hysteresis row.
- Caveat: For gas media at low gauge pressure the head row can be negligible; for oil it is often not.
Humidity
Humidity probe in a chamber against a chilled-mirror reference
A relative humidity probe is compared with a chilled-mirror hygrometer (dew point) and a reference thermometer in a climatic chamber.
Measurand: Correction to the UUT relative humidity at one point, in %rh.
| Contribution | Type · distribution | Illustrative value | Why this row and this distribution | Your number comes from |
|---|---|---|---|---|
| Repeatability of the comparison | A · s/√n (ν = 4) | 5 readings, u = 0.022 %rh | Standard deviation of the mean of repeated readings, s/√n, ν = n − 1 (GUM 4.2.3). Enter your readings; five to ten are usual. | Your repeated differences UUT − reference at the point. |
| Reference dew point certificate, as %rh | B · Normal, stated U and k | U = 0.3, k = 2 %rh | The reference’s calibration certificate states U and k; the standard uncertainty is U/k, normal (GUM 4.3.3–4.3.4). The dew-point uncertainty is converted to relative humidity at the chamber temperature (the humidity tool gives the sensitivity). | Dew-point U from the certificate × ∂RH/∂t_d at your point. |
| Reference temperature certificate, as %rh | B · Normal, stated U and k | U = 0.15, k = 2 %rh | Relative humidity depends on the air temperature at the UUT; the reference thermometer’s U is converted with ∂RH/∂t (about −0.3 %rh per 0.1 °C at 50 %rh, 23 °C). | Temperature U × ∂RH/∂t at your point. |
| Chamber uniformity (reference to UUT position) | B · Rectangular (uniform) | ± 0.4 %rh | Difference between where the reference and the UUT sit, bounded by the surveyed gradient, rectangular (GUM 4.3.7). | From your chamber survey, in %rh, including the temperature gradient converted to %rh. |
| Chamber stability during the readings | B · U-shaped (arcsine) | ± 0.3 %rh | A controlled medium cycles between limits around its set point; the arc sine (U-shaped) distribution fits a value that spends most time near the extremes (JCGM 101 6.4.6). | Half the peak-to-peak variation of the reference over the reading period. |
| UUT resolution | B · Rectangular (uniform) | ± 0.05 %rh | Half the least significant digit, rectangular (GUM F.2.2.1): the true value lies anywhere within the displayed step. | Half the UUT display step. |
| Reference drift since calibration | B · Rectangular (uniform) | ± 0.1 %rh | Change of the reference since its calibration, taken as equally likely anywhere within the bound, rectangular (GUM 4.3.7). | Your history for the mirror hygrometer and the thermometer. |
With these illustrative values the GUM budget gives u_c = 0.362 %rh, U = 0.724 %rh (k = 2, 95.45 %). Yours will differ.
Sources and caveats
- JCGM 100:2008 (GUM): distributions per the clauses named in each row.
- Sensitivities ∂RH/∂t_d and ∂RH/∂t from the site’s humidity tool (Hardy 1998 ITS-90 formulation).
- Structure of relative humidity calibration in a chamber as described in the free EURAMET calibration guide for humidity (euramet.org); no table reproduced.
- Caveat: Hysteresis of polymer RH sensors can be several tenths of %rh; add a row when the probe is calibrated rising and falling.
- Caveat: Convert everything to %rh at the chamber temperature before combining.
Starting structures only. Distributions per JCGM 100:2008 (4.2.3, 4.3.3–4.3.4, 4.3.7, 4.3.9, F.2.2.1, F.2.2.2, H.1) and JCGM 101:2008 6.4.6. The lists of contributions follow the usual comparison methods and the free EURAMET calibration guides and OIML R 111-1 Annex C in structure; no published example or table is reproduced. Templates are also available to AI assistants through the MCP server tool budget_templates.