Neutron STEM Lab
Virtual engineering lab

Monte Carlo Uncertainty & Model Comparison

Propagate input distributions with a reproducible seed, compare output statistics with first-order uncertainty, and interpret empirical coverage intervals.

Measurement / Engineering StatisticsadvancedValidated 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

Analytical estimate
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Output evaluated at the input estimates.
Analytical standard uncertainty
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First-order result for comparison.
Monte Carlo output mean
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Mean of completed output samples.
Monte Carlo output standard deviation
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Sample standard deviation of completed outputs.
Theory

Equations and model

Propagate distributions

Each Monte Carlo trial samples the stated input distributions, evaluates the measurement equation, and records one output. The output mean and sample standard deviation estimate the propagated distribution. The central coverage interval uses linearly interpolated empirical quantiles at 2.5% and 97.5%.

Yᵏ = f(X1ᵏ, X2ᵏ, …)MC mean = ΣYᵏ/MMC standard deviation = √(Σ(Yᵏ − mean)²/(M − 1))

Separate three uncertainty statements

The output standard deviation estimates physical uncertainty under the model. The empirical percentile interval describes output coverage. The standard error of the Monte Carlo mean describes finite simulation error; more simulated samples reduce this numerical error, not the physical input uncertainties. Seeds make comparison reproducible, while sample count still controls finite-run stability.

Simulation SEM = output SD/√MCentral 95% coverage interval = [Q0.025(Y), Q0.975(Y)]Normal denominator support near zero can make ratio moments undefined
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
—
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

  • Distinguish measurement uncertainty from Monte Carlo simulation error.
  • Compare analytical and sampled uncertainty for linear and nonlinear models.
  • Explain distribution support, coverage intervals, and repeatability.
Check your understanding

Questions to answer from the experiment

  1. What does increasing Monte Carlo sample count primarily reduce?
  2. What does the central 95% Monte Carlo interval summarize?
  3. Why does this tool reject an uncertain normal denominator for ratio Monte Carlo?
  4. What is needed to reproduce a Monte Carlo run?
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.