1 Foundations
1.1 What “propagation of uncertainty” means
Propagation of uncertainty refers to a collection of techniques for determining how uncertainty in measured input quantities influences the uncertainty in a computed output. The setting is usually a result obtained by evaluating a function of variables, where each variable is measured with a known spread (for example, a standard deviation) or with uncertainty bounds. Rather than treating the computed output as exact, the method produces an uncertainty estimate that reflects both the measurement noise and the way the calculation combines the inputs.
1.2 Types of uncertainty (random vs systematic)
Uncertainty is often described as having at least two broad components. Random uncertainty arises from variability that changes from one measurement to the next, such as fluctuations in instruments or environment, and is frequently modeled with a probability distribution. Systematic uncertainty represents effects that shift measurements in a consistent direction, such as calibration offsets, drift, or imperfect modeling assumptions. In practice, both kinds contribute to the overall uncertainty, though they are handled differently in some workflows.
1.3 Uncertainty measures and notation
Common uncertainty measures include standard uncertainty (often the standard deviation of an estimated quantity), variance, and expanded uncertainty (a scaled form used for reporting). Notation varies across fields, but a typical structure is to denote an input by \(x\) with uncertainty \(u(x)\), and an output by \(y=f(x_1,\dots,x_n)\) with uncertainty \(u(y)\). For distributional descriptions, the framework may also reference confidence levels or coverage factors.
1.4 Assumptions used in uncertainty analysis
Many analytic propagation formulas rely on assumptions about the input uncertainties. A central assumption is that the uncertainties are sufficiently small that a local approximation of the function is meaningful. Another frequent assumption is that the probability model used to represent uncertainty (e.g., normality) matches the measurement process closely enough. Additionally, independence between inputs is sometimes assumed unless covariance terms are provided.
1.5 Correlations and covariance basics
When two measured inputs vary together, their uncertainties are correlated. Correlation structure is captured by covariance, which measures how deviations in one variable relate to deviations in another. If the covariance is zero, inputs are often treated as independent in the uncertainty calculation. Nonzero covariance can increase or decrease the final uncertainty depending on the sign and the sensitivity of the output to each input.
2 Mathematical framework
2.1 Uncertainty of a function of variables
Consider an output \(y=f(x_1,\ldots,x_n)\), where each input \(x_i\) has uncertainty. The goal is to determine an uncertainty measure for \(y\) based on the uncertainties of the \(x_i\). For many practical cases, the uncertainty of \(y\) can be approximated by tracking how small perturbations in inputs change the value of \(f\). This leads naturally to approaches based on derivatives and covariance.
2.2 Linearization (Taylor expansion) method
A widely used analytic method is linearization: approximate \(f\) by a first-order Taylor expansion around nominal input values. Under this approximation, small changes in the inputs produce changes in the output that are linear combinations of the input deviations, enabling computation of the output variance from input variances and covariances.
2.2.1 Derivatives and sensitivity coefficients
Sensitivity coefficients quantify how responsive the output is to each input. In the linearization approach, the coefficient for variable \(x_i\) is the partial derivative \(\partial f/\partial x_i\) evaluated at nominal values. These derivatives weight the input uncertainties: even if an input has modest uncertainty, a large sensitivity can make it a dominant contributor to output uncertainty.
2.2.2 Covariance-matrix form
The linearized uncertainty can be expressed compactly using the covariance matrix \(\Sigma\) of the input vector \(\mathbf{x}\). Let the gradient of the function be \(\nabla f\). Then the approximate variance of \(y\) is \[ u^2(y)\approx \nabla f^\top \Sigma \nabla f. \] This formulation highlights how both variances (diagonal elements of \(\Sigma\)) and covariances (off-diagonal elements) enter the result.
2.3 Using partial derivatives in practice
In practice, derivatives may be computed symbolically, numerically, or derived from the structure of the measurement model. Numerical differentiation can be useful when the function is complex, but it introduces its own approximation errors. Analysts typically verify that derivative estimates are stable with respect to step size and that the resulting uncertainty does not change materially under reasonable numerical perturbations.
2.4 Handling units and dimensional consistency
A consistent uncertainty calculation must respect units. Sensitivity coefficients carry reciprocal units of the corresponding inputs so that the product of a derivative and a variable perturbation yields the output’s unit. Using a covariance-matrix approach can help ensure correct dimensional behavior because all terms ultimately combine to produce a variance with the correct physical units.
3 Special cases and common operations
3.1 Addition and subtraction
For an output \(y=x_1\pm x_2\), linear propagation yields \[ u^2(y)=u^2(x_1)+u^2(x_2)\pm 2\,\mathrm{cov}(x_1,x_2). \] If the inputs are treated as independent, the covariance term drops out, and the uncertainties add in quadrature. Correlations can either enlarge or reduce the net uncertainty, reflecting whether deviations reinforce or counteract each other.
3.2 Multiplication and division
For \(y=x_1x_2\) or \(y=x_1/x_2\), it is often convenient to work with relative uncertainties. Under linearization, the relative variance approximately combines as \[ \left(\frac{u(y)}{y}\right)^2 \approx \left(\frac{u(x_1)}{x_1}\right)^2+\left(\frac{u(x_2)}{x_2}\right)^2 \pm 2\frac{\mathrm{cov}(x_1,x_2)}{x_1x_2}, \] where the sign depends on whether the operation is multiplication or division. With independence, the cross term again vanishes.
3.3 Powers and roots
For power laws \(y=x^n\), propagation gives a simple scaling of uncertainty: \[
| u(y)\approx | n | \, | x | ^{n-1}u(x). |
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\]
| In terms of relative uncertainty, \(u(y)/y\approx | n | \,u(x)/x\) when the linear approximation is appropriate. For roots, \(y=\sqrt[m]{x}=x^{1/m}\), the factor becomes \(1/m\) in relative terms. |
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3.4 Logarithms and exponentials
| Nonlinear transformations can be handled via derivatives. For instance, if \(y=\ln x\), then \(\partial y/\partial x=1/x\), so \(u(y)\approx u(x)/ | x | \). For \(y=e^x\), the derivative is \(e^x\), leading to uncertainty scaling proportional to the function value itself. These results are frequently used when measurement models are linearized in a transformed variable. |
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3.5 Mixing constants with uncertain measurements
| If a calculation combines uncertain quantities with known constants, only the uncertain parts contribute to the propagated uncertainty. For example, \(y= a x + b\) with fixed \(a\) and \(b\) implies \(u(y)= | a | \,u(x)\). If the “constant” actually has uncertainty (e.g., a calibration factor), it should be included as an additional variable in the propagation to avoid underestimating output variability. |
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4 Correlated inputs
4.1 When independence is not valid
Independence may be inappropriate when inputs are derived from a common source, share systematic effects, or are produced by the same fitting procedure. For example, two parameters extracted from a common regression often have correlated estimation errors. Another situation occurs when measurement noise is filtered or processed in a way that couples channels, creating dependence between variables.
4.2 Covariance-driven propagation
With correlations, the covariance terms must be retained in the variance calculation. In the two-variable case, the cross term \(2\,\mathrm{cov}(x_1,x_2)\) reflects how the uncertainties move together. In the general multivariable setting, the covariance matrix \(\Sigma\) drives the output variance via \(\nabla f^\top \Sigma \nabla f\). This structure makes it clear that uncertainty propagation is not only about spreads but also about how those spreads interact.
4.3 Error budgets with correlation terms
An uncertainty budget typically lists contributors and how they combine into a total. When covariance exists, the budget may require additional “interaction” components rather than only summing individual contributions in quadrature. These interaction terms can be positive or negative, meaning that correlated inputs can either amplify or partially cancel in the final result.
4.4 Interpreting sign and magnitude of covariance effects
The sign of covariance has an intuitive impact: positive covariance tends to increase output uncertainty when the output increases with both inputs, while negative covariance can reduce it. However, the effect magnitude depends on the sensitivities as well. A large covariance might not matter if the output is weakly sensitive to one of the correlated variables, so interpretation should always reference the functional relationship.
5 Nonlinear and non-Gaussian situations
5.1 Limitations of linear propagation
Linearization assumes that the function is well approximated by its first-order behavior across the range of uncertainty. If the output depends strongly on inputs through curvature, the first-order approximation may misestimate variance. Additionally, if the true input distributions are wide or truncated, using a symmetric linear model can yield uncertainty estimates that do not match the actual distribution of the output.
5.2 Assessing whether linearization is adequate
Adequacy checks include comparing linearized results with higher-order expansions or with numerical propagation methods. Another approach is to examine whether second-order terms are small relative to first-order terms over the plausible input range. If the output uncertainty changes materially under reasonable modifications of the local region used for derivatives, linearization may be unreliable.
5.3 Transformations that change uncertainty shape
Some transformations map symmetric input uncertainty into skewed output uncertainty. For example, exponentials and logarithms can convert a near-Gaussian input into an output with non-Gaussian shape. Even when the goal is only a standard uncertainty, the underlying distributional shape influences whether variance-based approximations remain meaningful, particularly near physical bounds.
5.4 Bounds vs standard uncertainty
Standard uncertainty summarizes variability using a second-moment quantity, but some contexts require bounds or coverage intervals. When distributions are not approximately normal or when outputs are constrained (e.g., nonnegative quantities), it can be more appropriate to report interval estimates derived from quantiles. Bound-based reporting may better reflect asymmetry or truncation effects, though it depends on the chosen probabilistic model.
6 Numerical propagation methods
6.1 Monte Carlo simulation overview
Monte Carlo propagation estimates output uncertainty by repeatedly sampling input values from their uncertainty models and evaluating the function. The empirical variability of the resulting output samples approximates the distribution of \(y\). This method naturally handles nonlinear functions and can incorporate correlations through joint sampling using a covariance structure or other multivariate model.
6.2 Choosing input distributions
Inputs may be represented by normal, uniform, triangular, lognormal, or empirically derived distributions. The selection should reflect how uncertainty was obtained: instrumental noise often supports approximately normal models, while quantization or specification limits may motivate bounded uniform or other non-normal forms. For systematic effects, analysts may use models that represent plausible bias ranges, sometimes leading to asymmetric or non-Gaussian output distributions.
6.3 Sampling strategies and convergence checks
Convergence depends on the number of samples and on how stable the computed uncertainty metric is (e.g., standard deviation or a quantile interval). Variance reduction techniques can improve efficiency, but many applications rely on straightforward sampling with increasing sample counts until results stabilize. Convergence checks may include monitoring the running estimate of \(u(y)\), verifying that rare-tail events do not dominate, and ensuring that the results are insensitive to random seed changes within expected tolerance.
6.4 Reporting results from simulations
Monte Carlo workflows produce either an estimated distribution or summary metrics such as mean, standard deviation, and percentile-based intervals. Reporting typically includes the uncertainty measure used, the assumed input models, how many samples were drawn, and any burn-in or algorithmic details if relevant. When reporting expanded uncertainty, the analyst may use a coverage factor derived from the simulated distribution or rely on quantile definitions.
7 Constructing and interpreting uncertainty budgets
7.1 Identifying uncertainty sources
An uncertainty budget begins by listing sources that influence the measured inputs or the final computed result. These may include instrument resolution, calibration uncertainty, environmental effects, model-form uncertainty, and numerical computation tolerances. The analyst also decides whether uncertainty sources affect the inputs directly (as measurement noise) or indirectly (through parameters embedded in the model).
7.2 Classifying contributors (A-type, B-type if applicable)
Some standards classify contributors into measurement-type categories. A-type uncertainty is evaluated statistically from repeated observations, while B-type uncertainty is evaluated using other information such as manufacturer specifications, calibration certificates, or prior knowledge. This classification helps organize how each contributor is modeled probabilistically and how its distribution parameters are chosen.
7.3 Combining contributors into totals
Once individual contributions and their covariances (if any) are characterized, they are combined to form the total uncertainty. Under linear propagation, variances combine through derivative-weighted covariance expressions, which often reduce to quadrature sums under independence. If the analysis uses a numerical method, total uncertainty emerges from the simulated output variability, implicitly combining contributions according to the specified joint model.
7.4 Presenting results (expanded vs standard uncertainty)
Uncertainty reporting commonly distinguishes between standard uncertainty, which corresponds to a one-standard-deviation-type measure, and expanded uncertainty, which scales the standard uncertainty to achieve a desired coverage level. Expanded uncertainty may be reported alongside the associated coverage or degrees-of-freedom assumptions used to compute the scaling factor. Clear presentation also typically states the method used (analytic or simulation) and the key assumptions underpinning the result.
8 Validation, verification, and best practices
8.1 Checking reasonableness of propagated uncertainty
A practical check is whether the propagated uncertainty behaves sensibly when input uncertainties are reduced or when model sensitivity changes. For example, if increasing the uncertainty of an input leads to no change in output uncertainty, the formulation may be incorrect or that input may not actually influence the output. Analysts can also compare magnitudes against domain expectations, such as whether the final uncertainty is plausible relative to measurement resolution.
8.2 Comparing analytic vs simulation approaches
When possible, comparing analytic linearized results with Monte Carlo outputs is a strong validation step. Agreement supports the adequacy of linearization and the suitability of assumed distributions. Discrepancies often indicate strong nonlinearity, significant correlations not captured in the analytic model, or non-Gaussian effects that matter for the chosen uncertainty metric.
8.3 Robustness under modeling choices
Best practice includes sensitivity analysis on modeling choices: alternative input distributions, different treatments of systematic components, or alternative correlation assumptions. Robust results should not hinge on finely tuned assumptions. If output uncertainty varies widely across reasonable modeling choices, the analyst should report that dependency and consider collecting additional information to narrow uncertainty models.
8.4 Documentation and reproducibility of calculations
Uncertainty analysis is more credible when it is transparent. Documentation typically includes the function definition, nominal values, uncertainty models for each input, correlation assumptions, formulas or code used, and any numerical settings for simulation. Reproducibility is improved by using version-controlled scripts, recording random seeds for Monte Carlo runs when appropriate, and preserving input datasets used to estimate parameters.
9 Worked examples
9.1 Simple linear model example
| Suppose a measured variable \(x\) has standard uncertainty \(u(x)\), and the computed output is \(y=3x-2\). Since the derivative of \(y\) with respect to \(x\) is 3, linear propagation gives \(u(y)= | 3 | u(x)\). The constant term \(-2\) contributes no uncertainty because it is treated as exact. This example illustrates how, for linear functions, propagated uncertainty reduces to derivative scaling. |
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9.2 Function with multiple measured inputs
Let \(y=\sqrt{x_1x_2}\), where \(x_1\) and \(x_2\) are measured with uncertainties and are assumed independent. Using linearization, the partial derivatives are \(\partial y/\partial x_1=\tfrac{1}{2}\sqrt{x_2/x_1}\) and \(\partial y/\partial x_2=\tfrac{1}{2}\sqrt{x_1/x_2}\). The propagated variance becomes the sum of derivative-weighted variances: \[ u^2(y)\approx \left(\frac{\partial y}{\partial x_1}\right)^2u^2(x_1)+\left(\frac{\partial y}{\partial x_2}\right)^2u^2(x_2). \] This shows how both uncertainties contribute according to how strongly the function depends on each input.
9.3 Example with correlated measurements
Consider \(y=x_1-x_2\) where \(x_1\) and \(x_2\) have uncertainties and covariance \(\mathrm{cov}(x_1,x_2)\). The variance is \[ u^2(y)=u^2(x_1)+u^2(x_2)-2\,\mathrm{cov}(x_1,x_2). \] If the covariance is positive and the variables are subtracted, the output uncertainty can shrink because the correlated deviations partially cancel. If the covariance is negative, the same subtraction can increase uncertainty.
9.4 Example using Monte Carlo for a nonlinear function
Let \(y=\ln(x)\) with \(x\) measured such that it is positive and has uncertainty represented by a lognormal model or by a truncated distribution around a nominal value. A Monte Carlo method samples \(x\) repeatedly from the chosen distribution, computes \(y=\ln(x)\) each time, and estimates uncertainty from the resulting sample spread. This approach remains accurate even if the output distribution is skewed due to the logarithm, where linear variance propagation might not fully capture asymmetry.