BIZOMAT MAKEOS · Quality
Measurement Uncertainty Budget (Combined, Expanded, Test Ratio)
What is the total uncertainty of this length measurement, is it small enough against the tolerance, and what guard-banded limits should I accept to?
Uncertainty budget for a dimensional measurement: standard uncertainties from the gauge calibration certificate, the display resolution, repeatability, the Abbe error from an offset and tilt (as the Abbe error tool, taken as a rectangular limit), and the temperature uncertainty of the part, combined by root-sum-square, expanded with a coverage factor, the test uncertainty ratio against the tolerance band, and guard-banded acceptance limits inside the tolerance.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Tolerance band (total) | mm | Width of the tolerance band (upper minus lower limit). |
| Calibration uncertainty of the gauge (expanded, k = 2) | µm | Expanded uncertainty on the gauge’s calibration certificate (usually at k = 2). |
| Resolution of the gauge | µm | Smallest step of the gauge display. |
| Repeatability (standard deviation of repeat readings) | µm | Standard deviation of repeated readings on the same part (from a repeatability study). |
| Abbe offset | mm | Distance between the measuring line and the scale (Abbe offset). |
| Tilt of the moving element | arcsec | Tilt of the moving jaw or slide in arc-seconds. |
| Length measured | mm | Length being measured. |
| Part expansion coefficient | µm/m·K | Expansion coefficient of the part (the gauge is assumed corrected). |
| Uncertainty of the part temperature (±) | K | Plus-or-minus uncertainty of the part temperature from 20 °C after correction. |
| Coverage factor | Coverage factor for the expanded uncertainty (2 for about 95%). | |
| Least test uncertainty ratio | Least ratio of tolerance band to twice the expanded uncertainty you accept (often 4). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Tolerance band (total)
- What it is
- The full width of the tolerance band on the feature, from the lower limit to the upper limit.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.0001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Entered in millimetres while the contributors are in µm; use the total band, not the ± half.
What this means: Calibration uncertainty of the gauge (expanded,…
- What it is
- The expanded uncertainty stated on the gauge’s calibration certificate. The tool divides it by 2 to get a standard uncertainty.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- If the certificate uses a coverage factor other than 2, convert it first, because the tool assumes k = 2.
What this means: Resolution of the gauge
- What it is
- The smallest step the gauge display or scale can show.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It adds resolution ÷ √12 to the budget as a standard uncertainty.
What this means: Repeatability (standard deviation of repeat rea…
- What it is
- The standard deviation of repeated readings of the same feature with the same gauge and operator, from a repeatability study.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the standard deviation, not the range of the readings.
What this means: Abbe offset
- What it is
- The perpendicular distance between the line of measurement on the part and the gauge’s scale axis: the Abbe offset.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The Abbe error rises in direct proportion.
What this means: Tilt of the moving element
- What it is
- The angular tilt of the moving jaw or slide over its travel, which multiplied by the offset gives the Abbe error.
- Units and format
- Enter a number in arcsec: arcsec. The form accepts 0 arcsec to 3600 arcsec. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 arcsec as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- In arc-seconds: 3600 arc-seconds make one degree.
What this means: Length measured
- What it is
- The length of the feature being measured, used for the thermal expansion contribution.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The temperature contribution rises in direct proportion.
What this means: Part expansion coefficient
- What it is
- The coefficient of thermal expansion of the part material, from material data.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0 µm/m·K to 50 µm/m·K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 11.5 µm/m·K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- µm per metre per kelvin is the same number as ×10⁻⁶ per kelvin.
What this means: Uncertainty of the part temperature (±)
- What it is
- How far the part temperature may still be from the 20 °C reference after any correction, as a plus-or-minus limit.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0 K to 50 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Coverage factor
- What it is
- The multiplier applied to the combined standard uncertainty to give the expanded uncertainty at the confidence level you report.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The expanded uncertainty rises in proportion, lowering the test uncertainty ratio.
What this means: Least test uncertainty ratio
- What it is
- The smallest test uncertainty ratio you accept: tolerance band divided by twice the expanded uncertainty.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Calibration u = U ÷ 2; resolution u = res ÷ √12; repeatability u = s.
- Abbe limit a = offset × tan(tilt) (as the Abbe error tool), u = a ÷ √3; temperature limit a = L α ΔT, u = a ÷ √3.
- Combined u_c = √Σu²; expanded U = k u_c; TUR = tolerance ÷ 2U; guard-banded acceptance = tolerance limits brought in by U.
- Example: U = 5.9696 µm (k = 2); the test uncertainty ratio is 4.1879; accept within ±19.03 µm of nominal.
Limitations
- Uncorrelated contributors; correlated ones (same gauge, same temperature) add linearly.
- Gauge temperature effects assumed corrected; add them if not.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a gauge is good enough for a tolerance
- Set guard-banded acceptance limits
- Find the biggest contributor to measurement error
Who it is for
- Quality engineers
- Metrologists
- Inspectors
What you get
- Standard uncertainty of each contributor
- Combined and expanded uncertainty
- Test uncertainty ratio and guard-banded limits
Interpreting the result
- The largest contributor dominates the root-sum-square: reduce it first.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 709–735. Topic reference; no source tables or recommended settings are reproduced.
- ISO/IEC Guide 98-3 Uncertainty of measurement (GUM)
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Explains the concept · BIZOMAT original book
Uncertainty and calibration
In Flow Measurement by the Numbers
What it covers: Reading an accuracy specification, percentage of reading against span, zero stability at low flow, an orifice uncertainty budget, calibration and in-place checks, and a meter quoted in full scale.
Explains the concept · BIZOMAT original book
Setting the reject limit
In Production Leak Testing by the Numbers
What it covers: Turning a leak specification into a station threshold: gas conversion, an uncertainty budget, a guard band and its false-reject cost, and why retesting until a part passes defeats the limit.
Method explained here · BIZOMAT original book
Readings near the limits
In Accept or Reject by the Numbers
What it covers: What a reading near a tolerance limit says about true size: the uncertainty budget tool, test uncertainty ratio, combining the reading with the process, and false accept and reject rates.
Method explained here · BIZOMAT original book
Thickness gauging and calibration
In Ultrasonic Testing by the Numbers
What it covers: Two-point velocity and zero calibration, the V-path error of dual-element probes, temperature correction for hot walls, a Type 1 gauge study and an uncertainty budget for wall readings.
Useful alongside · BIZOMAT original book
Hardness traverses and microhardness
In Microstructures by the Numbers
What it covers: Measuring a hardened case with a microhardness traverse, keeping indentations valid, finding the effective hardened depth, and how diagonal measurement uncertainty limits the result.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
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