Type A and Type B Expanded Uncertainty Calculator
A measurement uncertainty budget starts with a handful of components, each converted to a standard uncertainty and combined by root sum of squares. This calculator handles the three most common terms so you can see which one drives your result.
Your numbers
Readings used to compute the sample standard deviation.
Smallest displayed increment.
Your results
Simplified budget with three components, with no sensitivity coefficients and no effective degrees of freedom adjustment. A full budget under the GUM may need more terms such as temperature, fixturing and reference standard drift.
Expanded uncertainty vs number of readings
Sweeps Number of repeat readings from 2 to 50, holding your other inputs fixed.
Structure uncertainty data in your system
Optional: discuss keeping uncertainty budgets and certificates tied to equipment records.
Optional. The tool and every export work without this. No spam.
How to use it
Enter the number of repeat readings and their sample standard deviation, the instrument resolution and the expanded uncertainty from the calibration certificate along with its coverage factor. Pick the coverage factor for your own result, usually 2. The tool returns each standard uncertainty, the combined value and the expanded value. The share outputs show which component contributes most of the variance, which tells you where improvement effort pays.
- Use readings taken under the actual measurement conditions.
- Check the certificate for its stated k, it is not always 2.
- Add other terms such as temperature by extending the budget manually.
Method and formulas
Type A uncertainty comes from statistics: the standard deviation of the mean is s divided by the square root of n. Type B comes from other knowledge. Resolution is treated as a rectangular distribution with half-width equal to half the resolution, so the standard uncertainty is resolution divided by 2 root 3. A certificate value is divided by its coverage factor to return to a standard uncertainty. Components are squared, added and rooted to give the combined standard uncertainty, then multiplied by k.
Worked example
Ten readings with s of 0.0012 mm give 0.00038 mm. A 0.001 mm resolution gives 0.00029 mm. A certificate of 0.002 mm at k=2 gives 0.001 mm. Combined, the value is about 0.00112 mm, and the expanded uncertainty at k=2 is about 0.0022 mm. The calibration term makes up roughly 80 percent of the variance, so more readings would barely help. A better reference standard would.
Common mistakes
Do not add standard uncertainties linearly, they combine in quadrature when independent. Do not use the certificate expanded value as if it were a standard uncertainty. Do not use resolution as the uncertainty itself, and do not forget that a small number of readings has few degrees of freedom, which in a formal budget would justify a larger k than 2. Finally, remember that this is the uncertainty of the instrument in use, not the full process.
Frequently Asked Questions
Why divide resolution by 2 root 3?
Rounding to the display makes the true value equally likely anywhere within plus or minus half a resolution step. For a rectangular distribution, the standard deviation is the half-width divided by the square root of 3.
What is coverage factor k?
It multiplies the combined standard uncertainty to give an interval with a stated level of confidence. K of 2 gives about 95 percent for a normal distribution and is the common default in calibration.
Can I use this for a calibration lab scope?
It is a teaching and planning tool. Accredited labs need a full budget meeting ISO/IEC 17025 and the GUM, including all relevant influence quantities and sensitivity coefficients.
Why does uncertainty plateau as readings increase?
Only the Type A term shrinks with more readings. Resolution and calibration terms stay the same, so the combined value approaches a floor set by them.
If you maintain uncertainty budgets in spreadsheets across many gages, we can help structure them alongside your ERP equipment records.
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