BIZOMAT MAKEOS · Quality
Independent Dimensional Uncertainty Budget
How do these independent measurement contributions combine?
Combine repeatability, calibration, resolution and one bounded dimensional contribution into standard and expanded uncertainty.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Reported result | choice | Choose a sample mean or one individual observation. |
| Sample standard deviation | mm | Sample standard deviation from independent repeated measurements under the relevant conditions. |
| Independent observations in the sample | observations | Observations used to estimate the standard deviation; also the averaging count in mean mode. |
| Calibration expanded uncertainty | mm | Expanded uncertainty from the calibration certificate in millimetres at the relevant point. |
| Calibration coverage factor | × | Coverage factor stated on that certificate; do not assume every certificate uses 2. |
| Digital resolution increment | mm | Smallest digital increment, modelled as independent rounding over ±half an increment. |
| Other rectangular half-width | mm | Half-width of one independent rectangular contribution, already converted to equivalent dimensional error in millimetres. |
| Output coverage factor | × | Coverage factor justified by the uncertainty procedure and effective degrees of freedom. |
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: Reported result
- What it is
- Whether the reported measurement result will be the average of the repeated readings or a single reading. It decides whether repeatability is divided by the square root of the number of observations.
- Units and format
- Choose one of 2 options from the list: Mean of the entered observations; One observation.
- Example
- The form opens with “Mean of the entered observations” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- The mean option shrinks the repeatability contribution by the square root of the observation count.
- Get it right
- Choose one observation if, in practice, a part is measured once; the mean option then would understate the uncertainty.
What this means: Sample standard deviation
- What it is
- The spread of repeated measurements of the same feature, taken independently under the conditions that apply in use, calculated as a sample standard deviation.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.01 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Its contribution is the standard deviation itself, or divided by √n for a mean.
- Get it right
- Repeat the full measurement each time (re-fixture, re-zero); repeated readings without resetting understate the spread.
What this means: Independent observations in the sample
- What it is
- How many independent repeated measurements were used to calculate the standard deviation. In mean mode it is also how many readings are averaged for the result.
- Units and format
- Enter a number in observations: observations. The form accepts 2 observations to 100000 observations. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 observations as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- In mean mode a larger count reduces the repeatability contribution; in single mode it has no effect.
- Get it right
- Count only independent repeats; several readings from one setup do not each count.
What this means: Calibration expanded uncertainty
- What it is
- The expanded uncertainty stated on the calibration certificate of the instrument or standard used, at the size you are measuring.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.004 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It is divided by the certificate coverage factor to give the calibration standard uncertainty.
- Get it right
- Convert certificate values to millimetres; certificates often quote µm, and 4 µm is 0.004 mm.
What this means: Calibration coverage factor
- What it is
- The coverage factor k that the calibration certificate used to turn its standard uncertainty into the expanded uncertainty you entered.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts 0.001 × to 100 ×. 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
- A larger certificate factor makes the calibration contribution smaller.
- Get it right
- Read k from the same certificate; do not assume a value if the certificate states another.
What this means: Digital resolution increment
- What it is
- The smallest step the instrument display can show, for example the last digit of a digital caliper or micrometer. Rounding to this step adds uncertainty.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.01 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The resolution contribution is the step divided by √12.
- Get it right
- Enter the full display step, not half of it; the tool converts it with √12 as a rectangular distribution.
What this means: Other rectangular half-width
- What it is
- Half the width of one other independent error source with known limits, such as a temperature effect, already converted into an equivalent dimensional error.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.003 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Its standard uncertainty is the half-width divided by √3.
- Get it right
- Enter the half-width (±a), not the full range; enter 0 if there is no such contribution.
What this means: Output coverage factor
- What it is
- The coverage factor k used to expand the combined standard uncertainty into the reported expanded uncertainty, chosen by your uncertainty procedure.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts 0.001 × to 100 ×. 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
- Expanded uncertainty is directly proportional to this factor.
- Get it right
- Justify k from your procedure and the effective degrees of freedom; it is a separate choice from the certificate factor.
Method and formulas
- Repeatability u = s/√n for a mean or s for one observation. Calibration u = certificate expanded uncertainty divided by its coverage factor.
- Resolution u = increment/√12. Other rectangular u = half-width/√3. Combined u is the square root of the sum of the four squared contributions. Expanded U = k × combined u.
- Example: s = 0.01 mm, n = 10, calibration U = 0.004 mm at k = 2, resolution 0.01 mm and half-width 0.003 mm gives combined u ≈ 0.005033223 mm and U ≈ 0.010066446 mm at output k = 2.
Limitations
- All contributions must be in millimetres with unit sensitivity coefficients. Covariances and general propagation coefficients are not modelled.
- Do not count resolution twice if another contribution already includes it. Resolution is not reduced by averaging here.
- k = 2 does not automatically establish 95% coverage. Small samples, distributions, bias corrections and degrees of freedom need review.
- An all-zero budget is rejected. No conformity decision or accredited certificate is issued.
- Default values are worked examples, not recommended settings or material properties.
Typical uses
- measurement budget
- repeatability
- coverage factor
Who it is for
- Metrology teams
- Quality engineers
- Inspectors
What you get
- Combined standard uncertainty
- Expanded uncertainty
- Component standard uncertainties and variance shares
Interpreting the result
- Document the measurement model and independence evidence before using the combined result.
- Variance shares help prioritise improvement. Record the quantity, unit, coverage factor and uncertainty together.
References
- NIST TN 1297: Law of propagation of uncertainty — https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-appendix-law-propagation-uncertainty
- NIST TN 1297: Reporting uncertainty — https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-7-reporting-uncertainty
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 709–738. Topic reference; implementation and examples are original, and no source tables are reproduced.
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