Calculator Inputs
Example Data Table
| Report Label | Measured Value | Type A | Type B | Covariance | Coverage Basis | k | Expanded Uncertainty | Reported Interval |
|---|---|---|---|---|---|---|---|---|
| Gauge Block Length | 100.0000 mm | 0.3500 mm | 0.2000 mm | 0.0000 | Normal, 95.45% | 2.000 | 0.8062 mm | 99.1938 to 100.8062 mm |
| Voltage Reference | 5.0000 V | 0.0120 V | 0.0080 V | 0.0000 | Student t, 95% | 2.086 | 0.0300 V | 4.9700 to 5.0300 V |
Formula Used
Combined standard uncertainty is calculated from the variance model below:
uc = √(uA2 + uB2 + 2cov)
Expanded uncertainty is then calculated as:
U = k × uc
The reported interval becomes:
y ± U
Here, y is the measured value, uA is the Type A standard uncertainty, uB is the Type B standard uncertainty, cov is the covariance term, and k is the selected coverage factor.
How to Use This Calculator
- Enter the measured value and optional unit.
- Provide Type A and Type B standard uncertainty estimates.
- Enter covariance when uncertainty components are correlated.
- Select a coverage factor method: normal table, Student t, or manual k.
- Set effective degrees of freedom when using Student t.
- Choose decimal places for the displayed report.
- Press the calculate button to show the result above the form.
- Review the interval, contribution table, and chart.
- Download the summary as CSV or PDF for documentation.
Frequently Asked Questions
1. What is expanded uncertainty?
Expanded uncertainty is the uncertainty interval reported around a measured value after multiplying the combined standard uncertainty by a coverage factor. It helps communicate a practical range likely to contain the true value.
2. What is the difference between combined and expanded uncertainty?
Combined standard uncertainty is the root-sum uncertainty from contributing components. Expanded uncertainty is the combined value multiplied by a coverage factor, giving a wider reporting interval for decisions, certificates, and audits.
3. When should I use k = 2?
k = 2 is commonly used for an approximate 95.45% level under a normal distribution. It is popular in calibration and quality reporting, but it may not fit every degrees-of-freedom situation.
4. Can covariance be negative?
Yes. Negative covariance can reduce combined variance when two components move in opposite directions. However, entering an excessively negative value can create an invalid negative variance, which the calculator blocks.
5. Why do degrees of freedom matter?
Degrees of freedom affect the Student t coverage factor. Smaller values usually require a larger factor, which increases expanded uncertainty. This is important when uncertainty is estimated from limited repeated data.
6. What is relative expanded uncertainty?
Relative expanded uncertainty expresses expanded uncertainty as a percentage of the measured value. It helps compare uncertainty across measurements with different magnitudes or units and supports process capability reviews.
7. Does this replace a full uncertainty budget?
No. This calculator summarizes the main math, but a complete uncertainty budget still needs source identification, assumptions, traceability, distribution choices, and supporting evidence from your quality system.
8. What unit should I report?
Use the same unit as the measured value unless your procedure requires a converted unit. The calculator appends the unit to output values so your exported report remains clear and consistent.