The Central Idea
Suppose a model output is determined by uncertain inputs
\[ Y=f(X_1,X_2,\ldots,X_n). \]If the inputs vary or are not known exactly, then the output cannot usually be known exactly either. The uncertainty in the output depends on two things:
- how uncertain the inputs are; and
- how sensitive the function \(f\) is to changes in those inputs.
A small uncertainty in a highly influential input can matter more than a large uncertainty in an input to which the model is relatively insensitive. Error propagation therefore connects directly to derivatives, sensitivity analysis, probability distributions, and model structure.
Error and Uncertainty Are Not the Same Thing
The traditional phrase error propagation is widely used, but in many applications uncertainty propagation is the more precise description.
An error is the difference between a measured or estimated value and the true value. The true value may be unknown, so the actual error may also be unknown.
Uncertainty describes the range, spread, or probability distribution used to represent what is not known exactly.
Propagation methods normally begin with uncertainty descriptions for model inputs and determine the uncertainty implied for the output. They do not magically reveal the unknown true error.
A Simple Model
Suppose the area of a rectangle is modeled by
\[ A=LW, \]where the measured length and width are uncertain. If
\[ L=10.0\pm0.1 \]and
\[ W=5.0\pm0.1, \]then the nominal area is
\[ A=(10.0)(5.0)=50.0. \]The output uncertainty depends on how the two input uncertainties are being interpreted and whether they are independent, correlated, statistical, or conservative bounds. Different propagation rules answer different forms of the uncertainty question.
Absolute and Relative Uncertainty
If a quantity is reported as
\[ x=x_0\pm\Delta x, \]then \(\Delta x\) is an absolute uncertainty. It has the same units as \(x\).
The corresponding relative uncertainty is
\[ \frac{\Delta x}{|x_0|}, \]when \(x_0\neq0\). A percentage uncertainty is
\[ \frac{\Delta x}{|x_0|}\times100\%. \]Absolute uncertainty is especially natural for sums and differences. Relative uncertainty is often more informative for products, quotients, and powers.
Conservative Bound Propagation
One approach treats stated uncertainties as bounds and asks how far the result could move under the most unfavorable combination of permitted input changes.
For a sum
\[ z=x+y, \]a conservative absolute bound is
\[ \Delta z\leq\Delta x+\Delta y. \]For a difference
\[ z=x-y, \]the same bound applies:
\[ \Delta z\leq\Delta x+\Delta y. \]The subtraction sign does not imply that the uncertainties themselves subtract. Either input may deviate in the direction that increases the output error.
Statistical Propagation
If uncertainties are represented statistically by standard deviations, and the relevant input errors are independent, variances rather than standard deviations add for sums and differences.
If
\[ Z=X+Y, \]with independent \(X\) and \(Y\), then
\[ \sigma_Z^2=\sigma_X^2+\sigma_Y^2. \]Therefore,
\[ \sigma_Z=\sqrt{\sigma_X^2+\sigma_Y^2}. \]The same variance rule applies to
\[ Z=X-Y \]when the inputs are independent.
Sum Example
Suppose two independent measured lengths are
\[ X=12.0\text{ cm},\qquad \sigma_X=0.3\text{ cm} \]and
\[ Y=8.0\text{ cm},\qquad \sigma_Y=0.4\text{ cm}. \]For
\[ Z=X+Y, \]the nominal value is
\[ Z=20.0\text{ cm}. \]The propagated standard uncertainty is
\[ \sigma_Z =\sqrt{(0.3)^2+(0.4)^2} =\sqrt{0.25} =0.5\text{ cm}. \]Thus a standard-uncertainty representation is
\[ Z=20.0\text{ cm},\qquad \sigma_Z=0.5\text{ cm}. \]Why Independent Uncertainties Combine by Root-Sum-of-Squares
Independent random deviations do not ordinarily move in the same direction at the same time. Positive and negative deviations can offset one another across repeated realizations.
That is why statistical uncertainty is often smaller than the conservative worst-case bound. For the previous example, the conservative sum of magnitudes would be
\[ 0.3+0.4=0.7\text{ cm}, \]while the independent standard uncertainty is
\[ \sqrt{0.3^2+0.4^2}=0.5\text{ cm}. \]These values answer different questions. A modeler should not switch between them without stating the interpretation being used.
Products and Quotients
For independent inputs with relatively small uncertainties, a useful first-order rule for a product
\[ Z=XY \]is
\[ \left(\frac{\sigma_Z}{|Z|}\right)^2 \approx \left(\frac{\sigma_X}{|X|}\right)^2 + \left(\frac{\sigma_Y}{|Y|}\right)^2. \]For a quotient
\[ Z=\frac{X}{Y}, \]the same independent first-order relative-variance rule applies:
\[ \left(\frac{\sigma_Z}{|Z|}\right)^2 \approx \left(\frac{\sigma_X}{|X|}\right)^2 + \left(\frac{\sigma_Y}{|Y|}\right)^2. \]These formulas are approximations derived from local linearization. They are most reliable when uncertainties are small relative to the nominal input values and the model is not strongly nonlinear over the uncertain region.
Product Example
Return to the rectangle with
\[ L=10.0,\qquad \sigma_L=0.1 \]and
\[ W=5.0,\qquad \sigma_W=0.1. \]The nominal area is
\[ A=LW=50.0. \]The relative standard uncertainty is approximately
\[ \frac{\sigma_A}{A} \approx \sqrt{ \left(\frac{0.1}{10.0}\right)^2+ \left(\frac{0.1}{5.0}\right)^2 }. \]Therefore,
\[ \frac{\sigma_A}{A} \approx \sqrt{0.01^2+0.02^2} \approx0.02236. \]Hence
\[ \sigma_A\approx50(0.02236)\approx1.12. \]The area may therefore be summarized, at the standard-uncertainty level, as
\[ A=50.0,\qquad \sigma_A\approx1.12. \]Powers
For a model of the form
\[ Y=X^n, \]a first-order approximation gives
\[ \frac{\sigma_Y}{|Y|} \approx |n|\frac{\sigma_X}{|X|}, \]provided the relative uncertainty is small and the nominal value is in a region where the expression behaves regularly.
Thus squaring approximately doubles relative uncertainty, while cubing approximately triples it.
Power Example
Suppose the radius of a sphere is measured as
\[ r=4.00\text{ cm},\qquad \sigma_r=0.04\text{ cm}. \]The volume is
\[ V=\frac{4}{3}\pi r^3. \]The relative uncertainty in \(r\) is
\[ \frac{\sigma_r}{r}=\frac{0.04}{4.00}=0.01. \]Because the volume depends on \(r^3\), the first-order relative uncertainty is approximately
\[ \frac{\sigma_V}{V}\approx3(0.01)=0.03. \]A one-percent standard uncertainty in radius therefore produces about a three-percent standard uncertainty in volume under this approximation.
The Derivative Method
The familiar propagation rules for sums, products, quotients, and powers are special cases of a more general idea based on derivatives.
For a one-variable model
\[ Y=f(X), \]a small input change \(\Delta X\) produces approximately
\[ \Delta Y\approx f'(X)\Delta X. \]If uncertainty is represented by a standard deviation, the corresponding first-order standard uncertainty is approximately
\[ \sigma_Y\approx|f'(\mu_X)|\sigma_X. \]The derivative acts as a local uncertainty multiplier. Large derivative magnitude means that small input uncertainty can create substantial output uncertainty.
Derivative Example
Suppose
\[ Y=X^2 \]and the uncertain input is centered near
\[ X=10,\qquad \sigma_X=0.2. \]Since
\[ f'(x)=2x, \]the local sensitivity at \(x=10\) is
\[ f'(10)=20. \]Therefore,
\[ \sigma_Y\approx20(0.2)=4. \]The nominal output is \(100\), so the relative uncertainty is about \(4\%\), consistent with the power rule: squaring approximately doubles the input's \(2\%\) relative uncertainty.
Multivariable Propagation
For a model
\[ Y=f(X_1,X_2,\ldots,X_n), \]the first-order change is approximated by the total differential
\[ dY \approx \frac{\partial f}{\partial x_1}dX_1 + \frac{\partial f}{\partial x_2}dX_2 +\cdots+ \frac{\partial f}{\partial x_n}dX_n. \]If the inputs are independent and uncertainty is represented by standard deviations, the first-order propagated variance is
\[ \sigma_Y^2 \approx \sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{X_i}^2. \]The partial derivatives are evaluated at the nominal or mean input values. Each term combines an input's uncertainty with the model's local sensitivity to that input.
Uncertainty Contributions
In the independent first-order formula, the contribution associated with input \(X_i\) is
\[ \left(\frac{\partial f}{\partial x_i}\right)^2\sigma_{X_i}^2. \]This reveals an important modeling distinction. An input may have a large standard uncertainty but contribute little to output uncertainty if the model is insensitive to it. Another input may be measured very precisely but still dominate output uncertainty because the model is highly sensitive to it.
Propagation analysis therefore helps identify where improved measurement, calibration, or parameter estimation would have the greatest effect on the reliability of the model output.
Correlation and Covariance
The independent-input formulas are not valid when input uncertainties are meaningfully related. If two uncertain quantities tend to move together, their covariance affects the propagated uncertainty.
For two inputs,
\[ Y=f(X_1,X_2), \]the first-order variance approximation is
\[ \sigma_Y^2 \approx \left(\frac{\partial f}{\partial x_1}\right)^2\sigma_1^2 + \left(\frac{\partial f}{\partial x_2}\right)^2\sigma_2^2 + 2\frac{\partial f}{\partial x_1} \frac{\partial f}{\partial x_2} \operatorname{Cov}(X_1,X_2). \]Positive covariance can increase output uncertainty when the two sensitivity terms have the same sign. Negative covariance can reduce it. If the sensitivities have opposite signs, the effect reverses.
Matrix Form of First-Order Propagation
For many uncertain inputs, covariance can be organized in a covariance matrix \(\Sigma\). Let
\[ \nabla f= \begin{bmatrix} \partial f/\partial x_1\\ \partial f/\partial x_2\\ \vdots\\ \partial f/\partial x_n \end{bmatrix}. \]Then the first-order output variance can be written compactly as
\[ \sigma_Y^2 \approx (\nabla f)^T\Sigma(\nabla f). \]This formula brings together multivariable calculus, linear algebra, and probability. It is one of the most reusable forms of local uncertainty propagation.
Vector-Valued Model Outputs
Some models produce several outputs at once:
\[ \mathbf{Y}=\mathbf{f}(\mathbf{X}). \]Near a nominal input, the model can be linearized using the Jacobian matrix \(J\). If the input covariance matrix is \(\Sigma_X\), then the propagated output covariance is approximately
\[ \Sigma_Y\approx J\Sigma_XJ^T. \]This form not only estimates uncertainty in each output but also describes how output uncertainties become correlated through the model.
Nonlinear Models
First-order propagation approximates a nonlinear model by its tangent line, tangent plane, or higher-dimensional linearization near a nominal point.
The approximation can become inaccurate when
- input uncertainty is large;
- the model is strongly curved over the uncertain region;
- the model contains thresholds or discontinuities;
- the input distribution is strongly skewed or heavy-tailed;
- the output is constrained by boundaries such as nonnegativity;
- the nominal point lies near a singularity or denominator near zero.
In such cases, directly propagating full probability distributions may be more appropriate than relying on a local derivative approximation.
Nonlinearity Can Shift the Mean
A nonlinear transformation can change not only the spread of an uncertain quantity but also the location of its expected output.
In general,
\[ E[f(X)]\neq f(E[X]). \]For example, if
\[ f(X)=X^2, \]then
\[ E[X^2]=\operatorname{Var}(X)+(E[X])^2. \]Evaluating the model only at the mean input gives \((E[X])^2\) and misses the variance contribution. This is one reason uncertainty analysis must consider the distribution of inputs rather than only a single representative value.
Propagation Through Repeated Calculations
Large models often contain many intermediate stages:
\[ X\longrightarrow U\longrightarrow V\longrightarrow Y. \]Uncertainty may be propagated stage by stage, but dependencies created in earlier stages must be preserved. Two intermediate quantities derived from the same uncertain source are generally not independent even if they appear as separate variables later in the calculation.
Ignoring those induced correlations can overstate or understate final uncertainty.
Why Dependence Matters: A Cancellation Example
Suppose
\[ U=X+1 \]and
\[ V=X-1. \]Then
\[ Y=U-V. \]Substitution gives
\[ Y=(X+1)-(X-1)=2. \]The uncertainty in \(X\) cancels exactly. If \(U\) and \(V\) were incorrectly treated as independent uncertain quantities, a propagation calculation would predict nonzero uncertainty in \(Y\). The mistake would come from ignoring their common dependence on \(X\).
Systematic Effects Do Not Average Away Automatically
Repeated measurements can reduce some random uncertainty, but repeated use of a biased measurement system can preserve the same systematic effect.
If every measurement is shifted by the same calibration error, averaging many measurements may produce a very precise estimate of the wrong value. Propagation analysis must therefore distinguish uncertainty components that behave independently from components that share a common source.
Dimensional Checks
Units provide a useful validation tool for propagation formulas.
If \(Y=f(X)\), then
\[ \sigma_Y\approx\left|\frac{dY}{dX}\right|\sigma_X. \]The derivative \(dY/dX\) has units of output per unit input. Multiplying by an input uncertainty therefore produces the correct output units.
Variances have squared units, while relative uncertainties and coefficients of variation are dimensionless. Unit inconsistency is often a warning that uncertainty terms have been combined incorrectly.
Reporting the Result
Propagation calculations may produce many decimal places, but the underlying uncertainty rarely justifies reporting all of them.
A common reporting practice is to round the uncertainty to a sensible number of significant digits and round the reported central value to the same decimal place.
For example, a calculated result such as
\[ 18.473826\pm0.362941 \]might reasonably be communicated as
\[ 18.47\pm0.36, \]depending on the conventions of the application. Excess digits can create a false impression of precision.
A Practical Propagation Workflow
- Write the model relationship clearly.
- Identify the uncertain inputs and parameters.
- Specify how each uncertainty is represented.
- Determine whether important inputs are independent or correlated.
- Choose a propagation method appropriate to the model and uncertainty size.
- Compute the nominal output.
- Propagate uncertainty using bounds, derivatives, covariance, simulation, or another justified method.
- Check units, constraints, and numerical reasonableness.
- Identify which inputs dominate the output uncertainty.
- Report assumptions and uncertainty interpretation with the result.
The propagation formula is only one part of the workflow. The validity of the result also depends on the quality of the input uncertainty model and the assumptions about dependence.
Common Mistakes
- Adding standard deviations directly: appropriate for some conservative bounds, but not for independent statistical uncertainties.
- Subtracting uncertainties: subtraction in the model does not generally mean uncertainty subtracts.
- Ignoring correlation: shared sources can materially change propagated uncertainty.
- Using relative uncertainty near zero: relative measures become unstable or undefined when the nominal value is near zero.
- Applying linear formulas to strongly nonlinear models: first-order approximations can fail over wide uncertainty ranges.
- Treating uncertainty as known error: uncertainty describes incomplete knowledge; it is not the hidden true error itself.
- Reporting too many digits: numerical output should not imply more precision than the model supports.
Error Budgets and Uncertainty Budgets
In engineering, experimental science, finance, operations, and other applied settings, modelers often organize major uncertainty sources into an uncertainty budget.
A budget may list each uncertain input, its estimated standard uncertainty, its sensitivity coefficient, and its contribution to the output variance. This makes the model auditable and helps prioritize improvement efforts.
| Input | Standard uncertainty | Sensitivity | Variance contribution |
|---|---|---|---|
| \(X_1\) | \(\sigma_1\) | \(\partial f/\partial x_1\) | \((\partial f/\partial x_1)^2\sigma_1^2\) |
| \(X_2\) | \(\sigma_2\) | \(\partial f/\partial x_2\) | \((\partial f/\partial x_2)^2\sigma_2^2\) |
When covariance is important, covariance terms must also be included rather than assigning all uncertainty independently to single inputs.
From Analytic Propagation to Simulation
Derivative-based propagation is efficient because it summarizes a model's local response with sensitivity coefficients. But not every uncertainty problem is well captured by a local linear approximation.
An alternative is to draw many plausible input combinations from their probability distributions, evaluate the model repeatedly, and study the resulting output distribution. That computational strategy is developed later in this section through Monte Carlo methods.
Analytic propagation and simulation are complementary tools. Simple models may yield transparent formulas, while complex models may benefit from direct numerical propagation.
Practice
Practice 1
Two independent measurements have standard uncertainties \(\sigma_X=0.6\) and \(\sigma_Y=0.8\). If \(Z=X+Y\), find \(\sigma_Z\).
Practice 2
Using the same values, find the propagated standard uncertainty if \(Z=X-Y\).
Practice 3
A quantity is measured as
\[ x=40.0\pm0.8. \]Find the relative uncertainty and percentage uncertainty.
Practice 4
Let
\[ Y=3X^2. \]If \(X=5.0\) and \(\sigma_X=0.1\), use the derivative method to estimate \(\sigma_Y\).
Practice 5
Two independent quantities have relative standard uncertainties of \(2\%\) and \(3\%\). If they are multiplied, estimate the relative standard uncertainty of the product.
Practice 6
A model is
\[ Y=4X_1-X_2. \]The inputs are independent with \(\sigma_1=0.5\) and \(\sigma_2=0.2\). Estimate \(\sigma_Y\).
Practice 7
Explain why two quantities derived from the same uncertain measurement should not automatically be treated as independent.
Practice 8
A nonlinear model has an input uncertainty large enough that the function changes curvature substantially across the plausible input range. Explain why a first-order derivative approximation may be unreliable.
Practice Solutions
Practice 1
For independent inputs,
\[ \sigma_Z =\sqrt{0.6^2+0.8^2} =\sqrt{0.36+0.64} =\boxed{1.0}. \]Practice 2
The independent variance rule is the same for a difference:
\[ \sigma_Z =\sqrt{0.6^2+0.8^2} =\boxed{1.0}. \]Practice 3
The relative uncertainty is
\[ \frac{0.8}{40.0}=0.02. \]Therefore, the percentage uncertainty is
\[ 0.02(100\%)=\boxed{2\%}. \]Practice 4
Since
\[ Y=3X^2, \]we have
\[ \frac{dY}{dX}=6X. \]At \(X=5\),
\[ \frac{dY}{dX}=30. \]Thus,
\[ \sigma_Y\approx30(0.1)=\boxed{3}. \]Practice 5
For an independent product,
\[ \frac{\sigma_Z}{|Z|} \approx \sqrt{0.02^2+0.03^2} =\sqrt{0.0013} \approx0.0361. \]The relative standard uncertainty is therefore approximately
\[ \boxed{3.6\%}. \]Practice 6
The sensitivities are
\[ \frac{\partial Y}{\partial X_1}=4, \qquad \frac{\partial Y}{\partial X_2}=-1. \]Therefore,
\[ \sigma_Y^2 \approx 4^2(0.5)^2+(-1)^2(0.2)^2. \] \[ \sigma_Y^2=16(0.25)+0.04=4.04. \]Hence,
\[ \sigma_Y\approx\sqrt{4.04}\approx\boxed{2.01}. \]Practice 7
A common uncertain source can cause the derived quantities to move together. Their covariance may therefore be nonzero. Treating them as independent can either exaggerate or understate the final uncertainty, especially when the model combines them by addition or subtraction.
Practice 8
First-order propagation replaces the nonlinear model locally with a linear approximation. If the uncertain region extends far enough for curvature to change the model response substantially, one local derivative no longer represents the whole region well. Higher-order methods or direct propagation of the input distributions may be needed.
What to Remember
- Error propagation describes how uncertain inputs create uncertainty in model outputs.
- Error and uncertainty are related concepts, but they are not identical.
- Conservative bounds and statistical standard uncertainties answer different questions.
- Independent variances add for sums and differences.
- Products, quotients, and powers are often analyzed naturally through relative uncertainty.
- Derivatives act as local sensitivity coefficients for uncertainty propagation.
- For independent inputs, first-order output variance is a weighted sum of input variances.
- Covariance must be included when input uncertainties are related.
- The matrix form \((\nabla f)^T\Sigma(\nabla f)\) generalizes first-order scalar propagation.
- Strong nonlinearity, large uncertainty, thresholds, and singularities can make linear propagation unreliable.
- Propagation analysis can identify which uncertain inputs dominate the reliability of a model.
Uncertainty does not simply attach itself to a final answer. It moves through the structure of the model. Understanding that structure is what makes error propagation a modeling tool rather than merely a collection of formulas.
Where We Go Next
Propagated uncertainty tells us how variable or uncertain a model output may be, but many applications require a more explicit interval statement about an estimated quantity or a future observation.
The next lesson, Confidence and Prediction Intervals, distinguishes intervals used to quantify uncertainty about estimated model quantities from intervals used to describe the range of plausible future outcomes.