Enter Measurement Data
Use repeated readings and Type B sources to build a full engineering uncertainty budget.
Example Data Table
This sample shows a typical dimensional inspection case with repeated readings and supporting Type B estimates.
| Measurement | Unit | Readings | Resolution | Calibration U | Calibration k | Drift | Environmental | Operator |
|---|---|---|---|---|---|---|---|---|
| Shaft Diameter | mm | 25.02, 25.00, 25.03, 24.98, 25.01, 25.04 | 0.01 | 0.02 | 2 | 0.015 | 0.010 | 0.008 |
Formula Used
1. Mean value: x̄ = Σx / n
2. Sample standard deviation: s = √( Σ(xᵢ - x̄)² / (n - 1) )
3. Type A standard uncertainty: uₐ = s / √n
4. Type B standard uncertainty: u = a / divisor, where the divisor is typically √3 for rectangular, √6 for triangular, or k for calibration certificates.
5. Combined standard uncertainty: u_c = √(uₐ² + u₁² + u₂² + ...)
6. Effective degrees of freedom: νeff = u_c⁴ / Σ(uᵢ⁴ / νᵢ)
7. Expanded uncertainty: U = k × u_c
8. Relative expanded uncertainty: (U / |x̄|) × 100
How to Use This Calculator
- Enter a clear measurement name and the working unit.
- Paste repeated readings separated by commas, spaces, or line breaks.
- Add instrument resolution and choose its assumed probability distribution.
- Enter calibration uncertainty from the certificate and its reported coverage factor.
- Add expected drift, environmental effects, and operator influence as Type B inputs.
- Select the reporting confidence level, then press the calculate button.
- Review the summary, expanded uncertainty statement, and source contribution table.
- Use the CSV or PDF buttons to export the result for reports.
Frequently Asked Questions
1. What does measurement uncertainty describe?
Measurement uncertainty describes the plausible spread around a reported value. It reflects repeatability, instrument limits, calibration information, and other influences that affect confidence in the result.
2. Why are repeated readings important?
Repeated readings quantify random variation. They provide the Type A component, which often reveals process repeatability better than a single observation.
3. What is the difference between standard and expanded uncertainty?
Standard uncertainty is the one-sigma combined estimate. Expanded uncertainty multiplies that value by a coverage factor, giving an interval linked to a chosen confidence level.
4. When should I use a rectangular distribution?
Use a rectangular distribution when any value inside fixed bounds is equally likely. Resolution limits and conservative drift bounds are common examples.
5. How is calibration uncertainty handled here?
Enter the uncertainty shown on the calibration certificate and the certificate coverage factor. The tool converts that expanded value into a standard uncertainty before combining it.
6. What does effective degrees of freedom mean?
Effective degrees of freedom estimate how much statistical confidence supports the combined result. Lower values usually increase the coverage factor needed for the same confidence.
7. Can I use this for dimensional and process measurements?
Yes. The calculator suits dimensional inspection, laboratory measurements, process checks, and other engineering applications where repeatability and known Type B sources must be combined.
8. What should I report in a final result?
Report the mean value, expanded uncertainty, unit, confidence level, and key assumptions. For audits, include the uncertainty budget and distributions used.