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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.

Load an example
ComponentValueDistributionkc_iν

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

Combined u_c
0.0448°C
root sum of squares
Expanded U (k = 2)
0.090°C
≈ 95 % coverage
ν effective
1461.4
Welch–Satterthwaite
Dominant
42 %
Chamber uniformity between reference and logger (±)
  1. Chamber uniformity between reference and logger (±)0.0289 · 42 %
  2. Logger resolution 0.1 °C (half-width)0.0289 · 42 %
  3. Logger repeatability (Type A, n = 10)0.0120 · 7 %
  4. Reference thermometer certificate (U, k = 2)0.01000 · 5 %
  5. Chamber stability during readings (Type A, n = 10)0.00800 · 3 %
  6. 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).

Get the budget as a Word document or PDF

Every component with its distribution, divisor, standard uncertainty, sensitivity and share of the variance, and the combined and expanded uncertainty, set out as a record.

The tool above stays free — this is only for the file

A worked example

A Pt100 is calibrated at 28 °C in a dry block against a reference thermometer. Five contributions are known, and the question is which of them actually decides the answer.

What you start with

Reference thermometer certificate
U = 0.030 °C at k = 2
Dry block axial uniformity
±0.050 °C, rectangular
Reference drift since its last calibration
±0.020 °C, rectangular
Indicator resolution
±0.005 °C, rectangular
Repeatability of 10 readings (Type A)
s = 0.012 °C

The arithmetic

  1. 01

    Divide each component by the divisor for its distribution.

    0.030 / 2 = 0.015 · 0.050 / √3 = 0.02887 · 0.020 / √3 = 0.01155 · 0.005 / √3 = 0.00289 · the Type A term is already a standard uncertainty at 0.012

  2. 02

    Square, add, take the root — the components are uncorrelated.

    u_c = √(0.015² + 0.02887² + 0.01155² + 0.00289² + 0.012²) = 0.0367 °C

  3. 03

    Expand at k = 2.

    U = 2 × 0.0367 = 0.073 °C

U = 0.073 °C at k = 2, approximately 95 % coverage.

The dry block supplies 62 % of the combined variance. The reference thermometer, which is the expensive item and the one people worry about, supplies 17 %. A better reference would move this budget by almost nothing; mapping the block, or immersing the sensor deeper, would move it a lot. A budget is worth building mainly because it tells you where not to spend money.

How the calculation works

  1. 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.
  2. 02Combined standard uncertainty for uncorrelated inputs: u_c² = Σ (c_i u_i)².
  3. 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.
  4. 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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Read the theory

Uncertainty

Building an uncertainty budget step by step

Write the model, list the inputs, evaluate each as a standard uncertainty, apply sensitivity coefficients, combine in quadrature, expand with k. One table.

2026-09-13 · 3 min read
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Type A vs Type B uncertainty: the difference that matters

Type A comes from repeated measurements, Type B from everything else. It changes nothing about how they combine, and everything about degrees of freedom.

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Rectangular, triangular and normal distributions in uncertainty budgets

The distribution sets the divisor that turns a limit into a standard uncertainty: √3 rectangular, √6 triangular, √2 U-shaped. Say which you chose.

2026-09-13 · 5 min read
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Effective degrees of freedom and when k = 2 is not enough

k = 2 assumes you know your uncertainty well. When a Type A term from three readings dominates, Welch-Satterthwaite puts k at 4.53 and more than doubles U.

2026-09-13 · 4 min read

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