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Measurement uncertainty budget calculator
Combine Type A and Type B components per the GUM: divisors by distribution, sensitivity coefficients, u_c, effective degrees of freedom and U at k = 2.
| Component | Value | Distribution | k | c_i | ν | |
|---|---|---|---|---|---|---|
Value column: for normal enter the expanded uncertainty and its k (certificates); for rectangular, triangular and U-shaped enter the half-width ±a (a 0.1 °C display is ±0.05); for standard enter a standard uncertainty directly (Type A) and its degrees of freedom.
Result
- Chamber uniformity between reference and logger (±)0.0289 · 42 %
- Logger resolution 0.1 °C (half-width)0.0289 · 42 %
- Logger repeatability (Type A, n = 10)0.0120 · 7 %
- Reference thermometer certificate (U, k = 2)0.01000 · 5 %
- Chamber stability during readings (Type A, n = 10)0.00800 · 3 %
- Reference drift since calibration (±)0.00577 · 2 %
Runs entirely in your browser · nothing you enter is sent to a server
Report U to two significant figures and the result to the same decimal place: u_c = 0.0448 → U = 0.090 °C (k = 2).
How the calculation works
- 01Standard uncertainty u_i = a_i / divisor: k for a normal (certificate) value, √3 rectangular, √6 triangular, √2 U-shaped, 1 for a value already expressed as a standard uncertainty.
- 02Combined standard uncertainty for uncorrelated inputs: u_c² = Σ (c_i u_i)².
- 03Effective degrees of freedom (Welch–Satterthwaite): ν_eff = u_c⁴ / Σ (c_i u_i)⁴ / ν_i, with ν = ∞ for Type B components without stated degrees of freedom.
- 04Expanded uncertainty U = k u_c; k = 2 by default, or from GUM Table G.2 at p = 95.45 % when automatic.
Limitations
- Assumes uncorrelated input quantities. Shared references or environments need covariance terms (see the article on correlated inputs).
- Automatic k uses the conservative table row at or below ν_eff; it does not interpolate.
- The budget is only as good as the component list — the tool cannot tell you what you left out.
Frequently asked questions
- What divisor do I use for a rectangular distribution?
- √3 ≈ 1.732. A limit of ±a becomes a standard uncertainty of a/√3. A digital display with resolution d has half-width d/2, giving d/(2√3).
- When should I use k other than 2?
- When a dominant Type A component rests on few observations. Enter its degrees of freedom (n − 1), switch k to automatic, and the tool applies GUM Table G.2.
- Is this the same calculation ValiTrac's engine performs?
- It is the same GUM arithmetic, implemented separately in your browser for public use. The platform's engineering engine records inputs, outputs and code version with an audit trail; this page does not store anything.
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Building an uncertainty budget step by step
Write the measurement model, list every input, evaluate each as a standard uncertainty, apply sensitivity coefficients, combine in quadrature, expand with k. Eight steps, one table.
Type A vs Type B uncertainty evaluation
Type A uses statistics on repeated observations; Type B uses any other information. Both produce standard uncertainties that combine the same way.
Rectangular, triangular and normal distributions in uncertainty budgets
The assumed distribution sets the divisor that turns a limit into a standard uncertainty: √3 for rectangular, √6 for triangular, k for normal.
Effective degrees of freedom and when k = 2 is not enough
When Type A components rest on few observations, the combined uncertainty has limited degrees of freedom and k must come from the t-distribution. Welch–Satterthwaite gives the number.