Calculator inputs
Formula used
1. Standard uncertainty for each source
u_i = a_i / d_i
Here, a_i is the entered estimate and d_i is the selected divisor.
2. Sensitivity-adjusted contribution
y_i = |c_i| × u_i
The sensitivity coefficient c_i converts each source into output units.
3. Combined standard uncertainty
u_c = √(Σ y_i²)
4. Expanded uncertainty
U = k × u_c
5. Relative expanded uncertainty
U_rel(%) = (U / |Y|) × 100
6. Effective degrees of freedom
ν_eff = u_c⁴ / Σ(y_i⁴ / ν_i)
This Welch–Satterthwaite estimate helps choose a coverage factor when some components have limited statistical support.
How to use this calculator
- Enter the measurand name, reported value, unit, and chosen coverage factor.
- Add one row for each uncertainty source such as repeatability, calibration, drift, resolution, or environment.
- Enter the source estimate. Use the divisor that matches how the estimate was obtained.
- Add the sensitivity coefficient for each source so every component is converted into the output quantity.
- Provide degrees of freedom when known. Use 999999 for effectively infinite values.
- Press Build uncertainty budget to calculate the combined and expanded uncertainty.
- Review the component table and Plotly chart to identify dominant contributors.
- Use the CSV and PDF buttons to export the current results for reports or audits.
Example data table
| Source | Type | Estimate | Divisor | Sensitivity | DoF | Comment |
|---|---|---|---|---|---|---|
| Reference standard | B | 0.010 V | 2 | 1.000 | ∞ | Certificate value converted from expanded uncertainty |
| Repeatability | A | 0.006 V | 1 | 1.000 | 9 | Standard deviation from repeated measurements |
| Resolution | B | 0.005 V | √3 | 1.000 | ∞ | Rectangular distribution from display resolution |
| Temperature effect | B | 0.008 V | √3 | 0.800 | 50 | Environmental estimate with partial sensitivity |
FAQs
1. What is an uncertainty budget?
An uncertainty budget is a structured list of all meaningful sources that affect a measurement result. It shows each source, its standard uncertainty, sensitivity coefficient, and final contribution to the combined uncertainty.
2. What is the difference between Type A and Type B?
Type A uncertainty comes from statistical analysis of repeated observations. Type B uncertainty comes from other information, such as calibration certificates, specifications, resolution limits, engineering judgment, or previous data.
3. Why do I need a divisor?
The divisor converts an entered estimate into a standard uncertainty. For example, rectangular distributions often use √3, triangular distributions use √6, and certificate values with k = 2 use 2.
4. What is a sensitivity coefficient?
A sensitivity coefficient describes how strongly one input influences the output quantity. It converts each input uncertainty into output units so different effects can be combined correctly.
5. What does the combined standard uncertainty represent?
It is the root sum square of all sensitivity-adjusted standard uncertainties. It represents the overall standard uncertainty of the final reported result before applying a coverage factor.
6. When should I use k = 2?
Many laboratories use k = 2 as a practical approximation for about 95% coverage when effective degrees of freedom are reasonably large. For smaller degrees of freedom, a different factor may be more appropriate.
7. Why are degrees of freedom included?
Degrees of freedom allow the calculator to estimate effective degrees of freedom with the Welch–Satterthwaite relation. That estimate helps justify a more suitable coverage factor for the final expanded uncertainty.
8. What does the Plotly graph show?
The chart displays the percentage share of total variance from each source. It quickly highlights dominant contributors, helping you focus on the components that most strongly affect measurement quality.