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VVUQ · Module 7 of 10

Uncertainty Propagation

Uncertain inputs make an uncertain output. Two methods carry the uncertainty through a model: the Taylor-series formula for smooth, mildly uncertain inputs, and Monte Carlo sampling for anything else.

Readiness check

This module carries uncertainty through a model. Tick only what you can do closed-notes.

Take a partial derivative of a function.
Combine independent uncertainties in quadrature.
Recall relative uncertainty as a fraction.
Recall a standard deviation and a standard error.
Recall that a random sample estimates a mean.

0 or 1 weak itemsContinue with this module.

2 weak itemsRevisit error propagation in Numerical Methods, Module 1.

3 or more weak itemsRevisit statistics in Mathematics, Module 15.

The core idea

Uncertainty propagation turns input uncertainties into an output uncertainty. The Taylor-series formula sums the squared sensitivities times input uncertainties; Monte Carlo samples the inputs and reads the output scatter, converging as one over the square root of the sample count.

u_y² = Σ (∂y/∂x_i)² u_{x_i}²Monte Carlo: sample inputs, read output spreadstandard error = s/√N

A model maps uncertain inputs to an output, and UQ must carry that input uncertainty through to the result. Two methods dominate. The Taylor-series (or law of propagation of uncertainty) method linearises the model about the nominal point: the output uncertainty is u_y = √(Σ (∂y/∂x_i)² u_{x_i}²), each input's uncertainty weighted by its sensitivity, combined in quadrature for independent inputs. For products and quotients this simplifies to adding relative uncertainties in quadrature. It is fast and analytic but assumes small, roughly Gaussian uncertainties and mild nonlinearity. Monte Carlo makes no such assumptions: it draws many random samples of the inputs from their distributions, runs the model on each, and reads the mean and spread of the output directly. It handles any distribution and any nonlinearity, at the cost of many model runs. Its estimate converges slowly, the standard error of the mean falling as s/√N, so halving the error requires four times the samples. The two methods are complementary: Taylor for quick, smooth cases, Monte Carlo for the rest.

The skill works when: you weight each input uncertainty by its sensitivity, or sample enough to converge the output spread.

The skill breaks down when: Taylor is used on a strongly nonlinear model, or Monte Carlo is stopped before it converges.

The concept. Uncertain inputs produce an uncertain output. Propagation finds the output spread, either analytically from sensitivities or numerically by sampling the inputs.

The skills, taught in order

Five skills build the Taylor formula, the Monte Carlo method, and how each converges.

7.1 The Taylor-series formula

Linearising the model gives u_y = √(Σ (∂y/∂x_i)² u_{x_i}²), the law of propagation of uncertainty. Each partial derivative is a sensitivity coefficient, so an input matters in proportion to both its own uncertainty and how strongly the output responds to it.

7.2 Products and relative uncertainty

For a product or quotient, the formula reduces to adding relative uncertainties in quadrature: (u_y/y)² = Σ (u_{x_i}/x_i)². This handy form covers many engineering formulas where quantities multiply.

Method	Assumes	Cost	Handles
Taylor series	small, smooth uncertainty	one derivative each	mild nonlinearity
Monte Carlo	nothing special	many model runs	any distribution or nonlinearity

Taylor is cheap but limited; Monte Carlo is general but costly. The choice depends on the model and the uncertainties.

7.3 The Monte Carlo method

Monte Carlo draws N random samples of the inputs from their distributions, evaluates the model on each, and takes the sample mean and standard deviation as the output estimate and uncertainty. It makes no linearity or normality assumptions and directly yields the output distribution.

7.4 Convergence of Monte Carlo

The standard error of the Monte Carlo mean is s/√N, so accuracy improves only as the square root of the sample count: cutting the error in half needs four times the runs. This slow convergence is the price of the method's generality.

7.5 Choosing a method

Use the Taylor formula when uncertainties are small, the model is smooth, and speed matters; use Monte Carlo when distributions are non-Gaussian, the model is strongly nonlinear, or the full output distribution is needed. Often the Taylor result guides how many Monte Carlo samples to run.

Engineering connection: the input uncertainty term in the validation uncertainty of Module 5 is produced by exactly this propagation.

Worked example 1: Taylor propagation for a product

An area is computed as y = L·W with L = 10 ± 0.1 and W = 5 ± 0.05 (standard uncertainties). Find the output value and its uncertainty.

Figure 1. Both inputs have a 1% relative uncertainty. Adding them in quadrature gives a 1.41% relative uncertainty on the area, or 0.71 in absolute terms.

ProblemFind the area and its uncertainty for the product in Figure 1.
Given / findy = L·W, L = 10 ± 0.1, W = 5 ± 0.05. Find y and u_y.
AssumptionsL and W are independent; uncertainties are small, so the linear formula applies.
ModelFor a product, (u_y/y)² = (u_L/L)² + (u_W/W)².
Equationsy = L·Wu_y/y = √((u_L/L)² + (u_W/W)²)
Solvey = 10 × 5 = 50. Relative: √(0.01² + 0.01²) = √0.0002 = 0.01414. u_y = 50 × 0.01414 = 0.71.
CheckEach input contributes a 1% relative uncertainty; combined in quadrature they give 1.41%, less than the 2% linear sum, as independent errors partly cancel.
ConclusionThe area is 50 ± 0.71. The relative uncertainties added in quadrature, the standard result for a product of independent quantities.

Result. y = 50, u_y = 0.71 (1.41% relative).

Worked example 2: Monte Carlo convergence

A Monte Carlo propagation with N = 400 samples gives an output sample standard deviation s = 1.5. Find the standard error of the mean and the 95% half-width, and say how many samples would halve the error.

Figure 2. The standard error falls as one over the square root of the sample count. Reaching half the error at N = 400 requires quadrupling to N = 1600.

ProblemFind the standard error, 95% half-width, and samples to halve the error in Figure 2.
Given / findN = 400, s = 1.5. Find SE, the 95% half-width, and N to halve SE.
AssumptionsIndependent samples; the mean is approximately normal for large N.
ModelSE = s/√N; 95% half-width ≈ 1.96·SE; SE scales as 1/√N.
EquationsSE = s/√Nhalve SE ⇒ 4× N
SolveSE = 1.5/√400 = 1.5/20 = 0.075. 95% half-width = 1.96 × 0.075 = 0.147. To halve SE to 0.0375, N must be 4 × 400 = 1600.
CheckAt N = 1600, SE = 1.5/40 = 0.0375, exactly half, confirming the one-over-square-root convergence.
ConclusionThe mean is known to ±0.075 (standard error), or ±0.147 at 95%. Cutting that in half costs four times the runs, the slow convergence of Monte Carlo.

Result. SE = 0.075, 95% half-width 0.147; N = 1600 to halve the error.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Taylor on a nonlinear model	Wrong output uncertainty	"Is the model roughly linear here?"	Use Monte Carlo for strong nonlinearity.
Ignoring sensitivity	All inputs weighted equally	"Did I include ∂y/∂x_i?"	Weight each input uncertainty by its sensitivity.
Too few Monte Carlo samples	Noisy, unconverged estimate	"Has the standard error settled?"	Increase N until SE is small enough.
Confusing s with SE	Uncertainty of the mean overstated	"Sample spread or spread of the mean?"	SE = s/√N is the uncertainty of the mean.

Practice ladder

Level 1 · Direct skill

For y = x² at x = 4 with u_x = 0.1, find u_y by the Taylor formula.

Show answer

∂y/∂x = 2x = 8, so u_y = |8| × 0.1 = 0.8.

Level 2 · Mixed concept

A Monte Carlo run with N = 100 gives s = 2.0. Find the standard error.

Show answer

SE = s/√N = 2.0/10 = 0.2.

Level 3 · Independent problem

For a quotient y = a/b with a = 20 ± 0.4 and b = 4 ± 0.04, find the relative uncertainty of y.

Show answer

(u_a/a)² + (u_b/b)² = 0.02² + 0.01² = 0.0005; relative u = √0.0005 = 0.0224 = 2.24%.

Transfer task | Real engineering

You must propagate uncertainty through a strongly nonlinear simulation. Argue for a method and how you would decide the sample count.

What good work looks like

Use Monte Carlo, since the Taylor formula fails for strong nonlinearity; choose N by increasing samples until the standard error s/√N is below the required precision, using an initial run to estimate s.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I weighted each input by its sensitivity in the Taylor formula."

"Give me three Monte Carlo runs; I will find the standard error of each."

"Propagate the uncertainty for me." Building the formula or sampling is the skill.

"How many samples do I need?" Reasoning from s/√N is the point.

Portfolio task

Propagate real input uncertainties through a model two ways, the Taylor formula and Monte Carlo, and compare the output uncertainties, noting where they agree.

Must include: a Taylor result, a Monte Carlo result with its standard error, and a comparison.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the Taylor propagation formula.

u_y = √(Σ (∂y/∂x_i)² u_{x_i}²).

2. How do relative uncertainties combine for a product?

In quadrature: (u_y/y)² = Σ (u_{x_i}/x_i)².

3. What does Monte Carlo do?

Samples the inputs, runs the model on each, and reads the output spread.

4. Write the Monte Carlo standard error.

SE = s/√N, the uncertainty of the mean.

5. How much does halving the error cost?

Four times the samples, since SE scales as 1/√N.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the Taylor and Monte Carlo results from a blank page.

+3 daysPropagate uncertainty through three models.

+7 daysRank the inputs by influence, Module 8.

+30 daysReuse propagation to produce the input uncertainty for a validation.

Standards mapping

This module follows the ASME Verification, Validation, and Uncertainty Quantification standards. Use these references to read further.

Topic in this module	Where to read more
Uncertainty propagation in validation	ASME V&V 20, CFD and Heat Transfer
Role of uncertainty quantification	ASME VVUQ 10.2, UQ in Solid Mechanics
Law of propagation of uncertainty	ISO Guide to the Expression of Uncertainty in Measurement (GUM)

Standard designations refer to the ASME V&V series; the Taylor propagation formula follows the ISO GUM, and Monte Carlo methods are standard across the UQ literature.