Uncertainty Budget Builder Calculator

Example data table

Formula used

• Standard uncertainty from limits: if a component is given as a half-width limit a, then u = a / divisor. Common divisors: rectangular √3, triangular √6, U-shaped √2.
• Combined standard uncertainty:
  u_c = √( Σ (c_i·u_i)² + 2ΣΣ (c_iu_i)(c_ju_j)ρ_ij )
  This calculator supports ρ(i,1) correlations with Component 1.
• Effective degrees of freedom:
  ν_eff = u_c⁴ / Σ( (c_iu_i)⁴ / ν_i )
• Expanded uncertainty: U = k·u_c, where k is chosen manually or estimated from ν_eff.

How to use this calculator

1. Enter a measurand label and its units, if available.
2. Add each uncertainty component as one row in the table.
3. Choose whether the magnitude is a standard uncertainty (u) or a half-width limit (a).
4. If you use a limit (a), select the distribution to convert it to u.
5. Enter sensitivity coefficients c_i from your measurement model.
6. Provide degrees of freedom ν_i for Type A terms when known.
7. Optionally add correlation with Component 1 using ρ(i,1).
8. Press “Build uncertainty budget” to see results above the form.
9. Use CSV and PDF buttons to export the budget table.

Article: building a defensible uncertainty budget

1) Why an uncertainty budget matters

An uncertainty budget turns a measurement into a decision-ready statement. It links your result to risk in pass/fail limits and process control. A strong budget lists meaningful sources, shows numerical contributions, and highlights which terms dominate the combined uncertainty.

2) Start with a measurement model

Define the measurand as a model, y = f(x₁, x₂, …). Each input x_i has an uncertainty u_i and a sensitivity coefficient c_i = ∂y/∂x_i. If units are consistent, c_i converts each input’s uncertainty into the measurand’s units.

3) Type A evaluation from repeated data

Type A components come from statistics on repeated observations. A common approach uses the sample standard deviation s, then the standard uncertainty of the mean is u = s/√n. Degrees of freedom are often ν = n − 1, which later influences the coverage factor for small datasets.

4) Type B evaluation from information sources

Type B components are built from certificates, instrument specifications, resolution limits, stability studies, and prior data. Many Type B terms arrive as limits ±a rather than standard uncertainties. Converting a to u depends on the assumed distribution, such as rectangular, triangular, or U-shaped.

5) Distributions and divisors with practical examples

Use a rectangular distribution when any value within ±a is equally likely; this gives u = a/√3. For a triangular distribution (more weight near zero), use u = a/√6. U-shaped gives u = a/√2; otherwise use a justified custom divisor.

6) Sensitivity coefficients and unit discipline

Sensitivity coefficients carry the physics. For direct readings, c_i is often 1. For a ratio, product, or calibration curve, c_i may differ from 1 and can amplify uncertainty. Always check that (c_i·u_i) has the measurand’s units; otherwise the budget becomes misleading.

7) Combining terms, correlation, and ν_eff

The combined variance adds squared contributions (c_iu_i)² and, when inputs are correlated, cross terms. Correlation can shift u_c when dominant terms share the same reference or environment. Welch–Satterthwaite ν_eff summarizes how limited Type A data affects reporting.

8) Reporting expanded uncertainty for stakeholders

Expanded uncertainty is U = k·u_c. A common reporting choice is 95% two-sided coverage; for large ν_eff, k is near 2. For smaller ν_eff, k can be higher and should be documented.

FAQs

1) What is the difference between Type A and Type B uncertainty?

Type A comes from statistical analysis of repeated measurements. Type B comes from other information sources like certificates, specifications, resolution limits, and prior studies, then converted into standard uncertainty terms.

2) How do I choose a distribution for a limit ±a?

Use rectangular when any value in the interval seems equally likely. Use triangular when values near zero are more likely. Use U-shaped when extremes are more likely. Document the choice based on evidence.

3) What does the sensitivity coefficient c_i represent?

c_i is how strongly the measurand changes with input x_i. In model terms, it is ∂y/∂x_i. It converts each input’s uncertainty into the measurand’s units.

4) How should I handle instrument resolution?

Resolution is commonly treated as a limit of ±(half the least significant digit). Many labs assume a rectangular distribution, so u = a/√3. If the device dithers or averages, justify a different model.

5) What does ν_eff tell me?

ν_eff is the effective degrees of freedom for the combined result. When Type A data are limited, ν_eff becomes small, which increases the coverage factor needed for the same confidence level.

6) When is it acceptable to use k = 2?

k = 2 is a common approximation for 95% coverage when ν_eff is large and the distribution is close to normal. For small ν_eff or strong non-normal behavior, estimate k from ν_eff.

7) How do I treat correlation between components?

If components share a common reference, environment, or calibration factor, correlation may exist. Enter a correlation coefficient when justified by analysis or evidence. Correlation adds cross terms that can raise or lower u_c.