Absolute Uncertainty Calculator — Error Propagation
Enter two measured values with their absolute uncertainties and choose an operation to find the combined absolute and relative uncertainty using standard in-quadrature propagation.
Operation
In-quadrature propagation assuming independent random uncertainties
- 1
δA²
0.5 × 0.5 = 0.25 - 2
δB²
0.3 × 0.3 = 0.09 - 3
Combined absolute uncertainty
√(0.25 + 0.09) = 0.5831In-quadrature rule for addition/subtraction: δZ = √(δA² + δB²).
How does this calculator work?
For A ± B: combined uncertainty δZ = √(δA² + δB²). For A × B or A ÷ B: δZ/|Z| = √((δA/|A|)² + (δB/|B|)²). Enter two values with their absolute uncertainties and choose an operation to propagate the uncertainty using the standard in-quadrature method.
Formula
How this is calculated
Absolute uncertainty (δA, in the same units as the measurement) quantifies how precise a measured value is. When two measured quantities are combined arithmetically, their uncertainties must be propagated to find the uncertainty of the result.
For addition and subtraction (Z = A ± B), the in-quadrature rule gives δZ = √(δA² + δB²). This quadrature sum assumes the two uncertainties are independent and random — it is smaller than the simple arithmetic sum δA + δB (which is used in worst-case or systematic analyses). For multiplication and division (Z = A × B or A ÷ B), the relative uncertainties combine: δZ / |Z| = √((δA/|A|)² + (δB/|B|)²), then δZ = |Z| × that relative value.
This calculator uses the in-quadrature (GUM-type) approach, standard in physics and engineering labs. If your course uses arithmetic sums instead, the result will be larger. Relative uncertainty (%) = δ/|value| × 100 and is useful for comparing precision across quantities with different units.
Frequently asked questions
Absolute uncertainty (δA) is expressed in the same units as the measurement (e.g. ±0.5 kg). Relative uncertainty is dimensionless: δA / |A| × 100 (e.g. ±1%). Use absolute when reporting a single measurement; use relative to compare precision between measurements of different magnitudes.
In-quadrature (√(δA² + δB²)) is appropriate for independent, random uncertainties — it gives the expected standard deviation of the result. Arithmetic sum (δA + δB) is used for worst-case or systematic bounds. Most physics and engineering standards (GUM) recommend in-quadrature.
For a digital instrument, take ±1 in the last displayed digit. For an analogue scale, take ±half the smallest graduation. If the uncertainty comes from repeated readings, use the standard deviation of the sample mean.
TG we-Calculate Editorial Team. (2026). Absolute Uncertainty Calculator — Error Propagation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/absolute-uncertainty-calculator
TG we-Calculate Editorial Team. "Absolute Uncertainty Calculator — Error Propagation." TG we-Calculate. 2026. https://we-calculate.com/calculator/absolute-uncertainty-calculator.
TG we-Calculate Editorial Team, "Absolute Uncertainty Calculator — Error Propagation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/absolute-uncertainty-calculator
@misc{wecalculate_absolute_uncertainty_calculator, title = {Absolute Uncertainty Calculator — Error Propagation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/absolute-uncertainty-calculator}}, year = {2026}, note = {TG we-Calculate} }
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