Neutron STEM Lab
Virtual engineering lab

Analytical Uncertainty Propagation

Follow exact sensitivity coefficients through engineering equations and inspect how input correlation changes combined and expanded uncertainty.

Measurement / Engineering StatisticsintermediateValidated educational model
Learning mode

Guided laboratory

Use the checklist when you want a structured lab. Explore mode leaves the instruments unrestricted.

1 · Define the measurement model

How do input uncertainties affect your result?

Enter each input estimate and its standard uncertainty u (one standard deviation). Unit changes preserve the physical values. Calculations use SI internally. General sum, difference, product, and ratio examples use unitless quantities.

Input correlations

ρ = 0 means uncorrelated inputs. Nonzero correlations require a consistent positive semidefinite correlation matrix. Analytical propagation accepts any supported marginals; correlated Monte Carlo currently supports jointly normal inputs.

Choose a model, check its inputs, then calculate or compare both methods.

The analytical method uses exact model derivatives in a first-order approximation. Monte Carlo propagates the stated distributions. For rectangular inputs, limits are estimate ±√3u; for symmetric triangular inputs, limits are estimate ±√6u. Normal inputs are unbounded.

Uncertain denominators must use bounded distributions that exclude zero for Monte Carlo. No invalid samples are silently discarded. k·u is an expanded uncertainty, and k = 2 does not guarantee exactly 95% coverage.

Keep this experiment

Setup stays on this device. Export JSON to move it to another browser.

Measurements

Instrument readings

Model estimate
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Output at the input estimates.
First-order standard uncertainty
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Combined standard uncertainty including covariance.
Expanded uncertainty
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First-order combined uncertainty multiplied by k.
Theory

Equations and model

Sensitivities connect input and output

A sensitivity coefficient is the partial derivative of the measurement equation with respect to one input, holding the others fixed. Its sign matters when correlated inputs are combined. Individual contributions are cᵢuᵢ in output units.

cᵢ = ∂f/∂xᵢuc² = Σ(cᵢuᵢ)² + 2Σᵢ<ⱼ cᵢcⱼuᵢuⱼρᵢⱼR = V/I; ∂R/∂V = 1/I; ∂R/∂I = −V/I²

Linearization is an approximation

The derivatives are exact for the selected equations, but replacing the whole model by its local tangent is an approximation. Strong curvature, broad distributions, or nearly singular denominators need special attention. A zero derivative can hide a nonzero second-order contribution. Monte Carlo helps explore these differences without certifying the physical model.

f(x + δx) ≈ f(x) + ΣcᵢδxᵢU = kuc
Validation

Independent checks

Model and covariance prerequisitesNot run

Finite estimates, nonnegative uncertainties, and a positive semidefinite correlation matrix.

Expected
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Simulated
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Error
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Tolerance
0%

Run the experiment to perform this check.

Required method completedNot run

Collect the analytical contributions, and Monte Carlo outputs when required by this experiment.

Expected
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Simulated
—
Error
—
Tolerance
0%

Run the experiment to perform this check.

Engineering interpretation

Run the experiment to generate an engineering interpretation.

Assumptions and limitations
Assumptions
  • Input uncertainties are standard uncertainties; units are converted consistently to SI before calculation.
  • Analytical sensitivities are exact derivatives evaluated at the input estimates; propagation is first order.
  • Monte Carlo output describes the selected measurement model and assumed input distributions. It does not validate those assumptions.
  • Correlated Monte Carlo uses jointly normal inputs; bounded rectangular/triangular distributions are independent.
Limitations
  • Monte Carlo uses 100–100,000 samples, yields every 1,000 samples, and supports cancellation. Percentile endpoint convergence is not automatically certified.
  • Uncertain normal denominators and bounded denominator distributions that include zero are rejected. No invalid outputs are silently dropped.
  • The central 95% output interval is empirical coverage under the model, not a confidence interval for the simulated mean.
  • Supported equations cover unitless sum/difference/product/ratio, resistance, resistor power, RC time constant, cylindrical density, and pendulum gravity. They do not include every physical non-ideality.
  • The pendulum model assumes small oscillations. Cylinder geometry and positive physical dimensions must be appropriate to the experiment.
Local experiment export

Save your measurements and setup

Exports are generated in your browser. No account or server upload is required.

Settings JSON
Current parameters and instrument controls, including waveform, output enable, scope coupling, timebase, trigger, and cursors when present. Data labs also include the dataset, mappings, exclusions, budget, or propagation setup.
Measurements CSV
Completed instrument readings with labels, units, and descriptions.
Plot PNG
The current canvas-based scope, transient, sweep, or statistics visualization when available.
Summary
Markdown report with objectives, setup, measurements, validation, interpretation, assumptions, and graded guided concept-check results.

Ready to export the current local experiment state.

Learning objectives

What this experiment should establish

  • Relate an input uncertainty to its signed sensitivity.
  • Compute independent and correlated uncertainty contributions.
  • Recognize when a first-order approximation can miss nonlinear effects.
Check your understanding

Questions to answer from the experiment

  1. For R = V/I, what is the sensitivity to current I?
  2. For z = x − y, equal input uncertainties, and perfect positive correlation, what can happen to the first-order uncertainty?
  3. What does analytical uncertainty propagation approximate here?
  4. Changing a resistance display from Ω to kΩ should do what?
Continue learning
Instrumentation and measurement errorGenerate repeated readings using the shared instrument error model, then transfer them into this lab.Measurement and uncertainty learning pathChoose another data analysis or uncertainty experiment.Mathematics toolsContinue with engineering mathematical methods and graphing.

Sources and model review

Reviewed 2026-09-16. The educational model exposes its assumptions and validation; source references do not imply external certification.