Calculator Input
Example Data Table
| Mode | Variable | Value | Uncertainty | Coefficient | Power |
|---|---|---|---|---|---|
| Linear | x1 | 12.5 | 0.3 | 2 | 1 |
| Linear | x2 | 8.2 | 0.2 | -1 | 1 |
| Product | x1 | 5.4 | 0.1 | 1 | 2 |
| Product | x2 | 3.1 | 0.05 | 1 | -1 |
Formula Used
Linear formula: y = c + Σ(cᵢxᵢ)
Linear propagated error: u(y) = √[Σ(cᵢuᵢ)² + 2ρΣ(cᵢuᵢ)(cⱼuⱼ)]
Product formula: y = s × Πxᵢᵃⁱ
Product propagated error: u(y) = |y|√[Σ(aᵢuᵢ/xᵢ)² + 2ρΣ(aᵢuᵢ/xᵢ)(aⱼuⱼ/xⱼ)]
Expanded uncertainty: U = k × u(y)
Reported interval: y − U to y + U
How to Use This Calculator
- Select linear mode for sums, differences, and weighted totals.
- Select product mode for multiplication, division, roots, and powers.
- Enter each measured value and its standard uncertainty.
- Use coefficients for linear weighted formulas.
- Use powers for product formulas. Use negative powers for division.
- Enter a common correlation only when variables are related.
- Set the coverage factor for the required confidence interval.
- Press calculate and review the propagated error above the form.
Propagation Error Formula Guide
What Propagation Error Means
Propagation error describes how measurement uncertainty moves through a formula. Every measured input has some variation. When those inputs combine, their uncertainties also combine. The final result should therefore include both an estimate and an uncertainty range. This calculator helps you make that report clearly.
Why It Matters in Statistics
Statistical work often depends on measured values. A mean, rate, ratio, index, or fitted result may use several uncertain inputs. Reporting only the final number can hide risk. A propagated error shows how stable that final number is. It also helps compare experiments, models, and quality checks.
Linear Calculations
Linear propagation is useful for addition, subtraction, and weighted sums. Each uncertainty is multiplied by its coefficient. The squared terms are then added. The square root gives the standard propagated error. If variables are correlated, the calculator can include a common correlation term.
Product and Power Calculations
Product propagation is useful for multiplication, division, powers, and roots. The calculator first finds the result. Then it combines relative uncertainty terms. A power increases or reduces the effect of each input. Negative powers represent division. Fractional powers represent roots.
Correlation and Confidence
Independent variables use zero correlation. Positive correlation can increase propagated uncertainty. Negative correlation can reduce it. Use correlation only when you have a defensible reason. The coverage factor expands the standard error into a wider interval. A common value is 1.96 for an approximate ninety five percent interval.
Interpreting the Output
The calculated value is the formula estimate. The standard propagated error shows one standard uncertainty. The expanded error multiplies that value by the coverage factor. The interval gives a practical lower and upper limit. The relative uncertainty shows the error as a percent of the estimate.
Good Reporting Practice
Always keep units consistent. Use standard uncertainties, not broad guesses. Avoid extra decimal places when the uncertainty is large. State the formula, the input values, and the coverage factor. This makes the result easier to review and repeat.
FAQs
What is propagation error?
Propagation error is the uncertainty transferred from input measurements into a calculated result. It shows how reliable the final formula output may be.
When should I use linear mode?
Use linear mode for addition, subtraction, weighted totals, adjusted scores, or formulas where each variable is multiplied by a coefficient.
When should I use product mode?
Use product mode for multiplication, division, square roots, squared terms, ratios, rates, and formulas that contain powers.
What does standard uncertainty mean?
Standard uncertainty is the estimated standard deviation of a measured value. It should be entered in the same unit as the value.
What is the coverage factor?
The coverage factor expands standard uncertainty into a wider interval. A value near 1.96 is often used for approximate ninety five percent coverage.
Should correlation always be entered?
No. Use zero when variables are independent. Enter correlation only when measurements are linked and you have a reasonable estimate.
How do I enter division in product mode?
Use a negative power. For example, dividing by x2 is entered by setting the x2 power to -1.
Why is relative uncertainty useful?
Relative uncertainty compares error with the final estimate. It helps judge whether the result is precise or too uncertain for decisions.