Uncertainty Propagation Calculator — GUM, Correlation & Expanded Uncertainty
Calculate combined standard and expanded uncertainty for sums, differences, products, quotients, powers, product ratios, or a general GUM model, with optional input correlations and coverage factor k.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Propagation model
Input quantities
Exponent
General GUM model
Correlation
Reported uncertainty
Results
Calculation tools Save, export and manage your results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
General law of propagation of uncertainty
For a result y = f(x₁, x₂, ...), the calculator uses first-order uncertainty propagation based on sensitivity coefficients:
uc(y) = √[Σ(ciui)² + 2Σ(cicjρijuiuj)]
where:
- uc(y) — combined standard uncertainty of the result,
- ui — standard uncertainty of input quantity i,
- ci = ∂f/∂xi — sensitivity coefficient,
- ρij — correlation coefficient between inputs i and j, from −1 to +1.
When the inputs are treated as independent, all correlation coefficients are set to zero and the covariance terms disappear.
Addition: y = a + b
ca = 1, cb = 1
uc² = u(a)² + u(b)² + 2ρabu(a)u(b)
Subtraction: y = a − b
ca = 1, cb = −1
uc² = u(a)² + u(b)² − 2ρabu(a)u(b)
Multiplication: y = a · b
ca = b, cb = a
uc² = [b·u(a)]² + [a·u(b)]² + 2abρabu(a)u(b)
Division: y = a / b
ca = 1/b, cb = −a/b²
uc² = [u(a)/b]² + [a·u(b)/b²]² + 2(1/b)(−a/b²)ρabu(a)u(b)
The denominator b must be non-zero.
Power: y = an
ca = n·an−1
uc(y) = |n·an−1|·u(a)
This mode uses a single uncertain input and therefore does not apply an input-correlation term.
Product ratio: y = (a · b) / c
ca = b/c, cb = a/c, cc = −ab/c²
The denominator c must be non-zero. Correlations ρab, ρac, and ρbc may be included.
General GUM model
In General GUM mode, the nominal result y is entered directly together with sensitivity coefficients and standard uncertainties:
uc²(y) = (cau(a))² + (cbu(b))² + (ccu(c))² + 2cacbρabu(a)u(b) + 2caccρacu(a)u(c) + 2cbccρbcu(b)u(c)
Inputs a and b are required. The third input c is optional, but if it is used, both its sensitivity coefficient and its standard uncertainty must be provided.
Expanded uncertainty
U = k · uc(y)
where k is the selected coverage factor.
Relative standard uncertainty
uc,r = uc(y) / |y| · 100%
Relative standard uncertainty is reported only when y ≠ 0.
Expanded interval
Lower bound = y − U
Upper bound = y + U
Uncertainty contribution for independent inputs
When covariance terms are not used, the calculator also estimates the variance contribution of each input:
Contributioni = (ciui)² / uc² · 100%
Contribution percentages and the dominant contributor are not assigned when correlated inputs are used because covariance terms form part of the combined variance.
When to use this calculator
Use this calculator when a calculated quantity depends on one or more measured or estimated inputs that already have known standard uncertainties, and you need to determine how those uncertainties propagate into the final result.
Typical applications include:
- propagating measurement uncertainty through sums and differences,
- calculating uncertainty in products and ratios,
- propagating uncertainty through a power relationship such as y = aⁿ,
- evaluating a three-variable relationship of the form y = (a·b)/c,
- including known correlation between measured input quantities,
- calculating combined standard uncertainty and expanded uncertainty,
- estimating relative uncertainty of a calculated result,
- identifying the dominant uncertainty contributor when inputs are independent,
- performing a first-order GUM-style calculation when sensitivity coefficients are already known.
The General GUM mode is useful when the measurement model is more complex than the predefined equations but the nominal result and partial-derivative sensitivity coefficients have already been determined elsewhere.
Correlation should be included when two or more input quantities are statistically dependent or share a common influence that makes their errors vary together. Ignoring significant correlation can either underestimate or overestimate the combined uncertainty.
Do not use this calculator as a substitute for a complete measurement uncertainty evaluation when you still need to determine Type A or Type B uncertainty components, probability distributions, effective degrees of freedom, systematic corrections, or an appropriate coverage factor.
The calculator uses first-order propagation. For strongly nonlinear measurement models, large input uncertainties, discontinuities, or distributions that become significantly non-normal after transformation, a distribution-propagation or Monte Carlo method may be more appropriate.
How to interpret the result
y — Result is the nominal output calculated from the selected equation, or the nominal output entered directly in General GUM mode.
uc(y) — Combined standard uncertainty represents the propagated standard uncertainty associated with the result after applying the sensitivity coefficients and, when selected, the entered correlations.
A larger value of uc means that the uncertainty of the inputs has a larger effect on the calculated output. Reducing the uncertainty of an input with a large sensitivity-weighted contribution will generally reduce the combined uncertainty most effectively.
U — Expanded uncertainty is calculated as:
U = k · uc
Increasing the selected coverage factor increases U and widens the displayed interval y ± U. The calculator does not determine whether the selected k corresponds to a particular confidence level.
Relative standard uncertainty expresses uc as a percentage of the magnitude of the nominal result. It is useful for comparing uncertainty levels between results of different magnitudes. It is not reported when y = 0.
Lower and upper bounds are calculated as y − U and y + U. They are arithmetic expanded-uncertainty bounds, not automatically a statistically validated confidence interval.
For independent inputs, the calculator also reports the percentage of propagated variance associated with each input and identifies the largest contributor. A high contribution indicates that the combination of that input's standard uncertainty and sensitivity coefficient dominates the independent-input uncertainty budget.
When correlated inputs are used, separate percentage contributions and a dominant contributor are not assigned because covariance terms contribute to the total variance.
Positive correlation does not always increase uncertainty and negative correlation does not always decrease it. The effect also depends on the signs of the sensitivity coefficients. For example, subtraction introduces a negative sensitivity coefficient for the second input.
A successfully calculated result does not represent a pass/fail or safety decision. This calculator evaluates uncertainty propagation only. Compliance with a specification, tolerance, measurement capability requirement or engineering acceptance criterion must be assessed separately.
Calculation example
Example: addition of two correlated measured quantities
Two measured quantities are added:
y = a + b
Assume:
- a = 100.0
- u(a) = 0.5
- b = 50.0
- u(b) = 0.3
- ρab = 0.50
- coverage factor k = 2
The nominal result is:
y = 100.0 + 50.0 = 150.0
For addition, both sensitivity coefficients are equal to 1:
ca = 1
cb = 1
The propagated variance including correlation is:
uc² = (1·0.5)² + (1·0.3)² + 2·1·1·0.50·0.5·0.3
uc² = 0.25 + 0.09 + 0.15 = 0.49
Therefore:
uc(y) = √0.49 = 0.70
With k = 2:
U = 2 · 0.70 = 1.40
The relative standard uncertainty is:
uc,r = 0.70 / 150.0 · 100% = 0.467%
The expanded interval reported by the calculator is:
Lower bound = 150.0 − 1.40 = 148.60
Upper bound = 150.0 + 1.40 = 151.40
The result can therefore be reported numerically as:
y = 150.0, uc = 0.70, U = 1.40 for k = 2
Because correlation was included, the calculator does not assign separate percentage contributions or a dominant contributor. The covariance term forms part of the total propagated variance and cannot be represented by the independent-input contribution percentages used by the calculator.
If the same inputs were incorrectly treated as independent, the calculated standard uncertainty would be √(0.25 + 0.09) ≈ 0.583 instead of 0.700. This illustrates why a significant known correlation should not be ignored.
The calculation is a first-order uncertainty propagation. The interval y ± U should not automatically be interpreted as a specific confidence interval unless the statistical assumptions supporting the selected coverage factor are also justified.
Assumptions and limitations
- The calculation uses first-order propagation of uncertainty based on local sensitivity coefficients evaluated at the nominal operating point.
- Input uncertainties must be supplied as standard uncertainties. A tolerance, maximum error, calibration limit or ± specification should not be entered directly unless it has already been converted to an appropriate standard uncertainty.
- The calculator does not independently evaluate Type A and Type B uncertainty components, probability distributions, instrument resolution, calibration corrections or repeated-measurement statistics.
- Standard uncertainties must be finite and greater than or equal to zero.
- Inputs are treated as independent by default. When dependence is significant, appropriate correlation coefficients should be entered.
- Every entered correlation coefficient must satisfy −1 ≤ ρ ≤ +1. For three-input models, the complete set of entered correlations must also form a positive-semidefinite correlation matrix.
- The predefined modes cover addition, subtraction, multiplication, division, a single-variable power, and the three-variable product ratio (a·b)/c. General GUM mode supports two required uncertainty inputs and one optional third input.
- General GUM mode assumes that the entered nominal result and sensitivity coefficients have been determined correctly outside the calculator. The calculator does not derive those coefficients from a user-defined equation.
- The denominator must be non-zero in division and product-ratio modes.
- For a negative base in power mode, the exponent must be an integer for the result to remain real-valued.
- First-order propagation at a = 0 is only accepted by the implemented power model in cases where the local derivative is well defined by the calculator's rules. Other zero-base combinations may be rejected.
- Relative standard uncertainty is undefined when the nominal result y equals zero.
- No automatic unit conversion or dimensional analysis is performed. Input quantities, standard uncertainties and sensitivity coefficients must use mutually consistent units.
- The calculator does not calculate effective degrees of freedom or automatically derive a coverage factor from a requested confidence probability.
- The expanded interval y ± U is produced directly from the selected coverage factor. It should not automatically be interpreted as a particular confidence or coverage probability without additional statistical justification.
- Independent-input contribution percentages are based only on squared sensitivity-weighted standard-uncertainty terms. They are intentionally not reported when covariance terms are present.
- Strong model nonlinearity, large uncertainties, asymmetric output distributions, discontinuities or other situations in which local linearization is inadequate may require a higher-order treatment or propagation of distributions such as a Monte Carlo method.
- The calculator supports a first-order GUM-style uncertainty propagation calculation but does not constitute a complete implementation of every procedure, reporting requirement or validation step defined in the GUM family of documents.