OpenSTEMTool / Data & Research / Statistics & Probability
Uncertainty Propagation – Free Online Scientific Calculator
Propagate uncertainties through common two-variable arithmetic functions including optional correlation.
Interactive & 100% Free: Enter your values directly in the calculator above or change the default inputs below to compute outputs instantly in your browser. All computations run locally for immediate privacy and high performance.
What this tool calculates
Calculate measurement error correctly. This computational model is built for university students, educators, lab researchers, and practicing engineers who need transparent, verified scientific calculations.
Calculation method and underlying formula
First-order propagation for f = x × y, using σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + 2ρ(∂f/∂x)(∂f/∂y)σxσy.
The calculation evaluates relationships between input variables according to standard scientific principles. Inspect the editable parameter table below to test specific values.
Step-by-step calculation guide
- Input Parameters: Enter the required values into the parameter table below.
- Model Evaluation: The system automatically evaluates the governing equations locally with high numerical precision.
- Analyze & Export: Review the calculated outputs, inspect the generated curve plot, and download publication-ready reports or CSV data.
Worked example: default input values
The default worked example demonstrates calculation with the following parameters. You can modify any value in the interactive panel to re-evaluate your specific scenario:
| Parameter | Example value | Unit |
|---|---|---|
| Formula | multiply | — |
| Value x | 12 | — |
| Standard uncertainty σx | 0.2 | — |
| Value y | 4 | — |
| Standard uncertainty σy | 0.1 | — |
| Correlation coefficient ρ | 0 | — |
| Power n | 2 | — |
Worked example results
Based on the default parameters entered above, the model computes the following quantities:
| Calculated quantity | Output value | Unit |
|---|---|---|
| Calculated f | 48 | — |
| Standard uncertainty | 1.44222 | — |
| Relative uncertainty (%) | 3.00463 | % |
| ∂f/∂x | 4 | — |
| ∂f/∂y | 12 | — |
Assumptions and interpretation
- First-order approximation is less reliable for large errors, strong nonlinearities, or distributions near singularities.
- Correlation is ignored for xⁿ because it only depends on x.
Educational calculations are not substitutes for professional engineering design approvals, certified laboratory assays, clinical diagnostics, or peer-reviewed scientific publications. Always inspect units, verify physical domains, and evaluate boundary conditions.
Frequently asked questions (FAQ)
How does the Uncertainty Propagation calculator work?
The calculator computes results locally in your browser using the established mathematical model: First-order propagation for f = x × y, using σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + 2ρ(∂f/∂x)(∂f/∂y)σxσy.
What are the input parameters for Uncertainty Propagation?
The required parameters include: Formula (dimensionless), Value x (dimensionless), Standard uncertainty σx (dimensionless), Value y (dimensionless), Standard uncertainty σy (dimensionless), Correlation coefficient ρ (dimensionless), Power n (dimensionless).
What assumptions apply to this Uncertainty Propagation model?
First-order approximation is less reliable for large errors, strong nonlinearities, or distributions near singularities.. Correlation is ignored for xⁿ because it only depends on x.
Can I export results from this calculator?
Yes. OpenSTEMTool supports exporting publication-ready SVG charts, PNG graphics, raw CSV data tables, JSON parameter archives, and printable HTML/PDF scientific reports.
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