Uncertainty & measurement
Uncertainty Propagation Calculator
Combine two measurements, or raise one to a power, using the common school-lab uncertainty rules.
Your measurements
Start with the example, or enter your own values. Use a dot for decimals and e for scientific notation.
Calculated locally. No account or uploads.
Result & working
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Step-by-step working
The formula
z = a ± b: Δz = Δa + Δb
z = ab or a/b: p_z ≈ p_a + p_b
z = aⁿ: p_z ≈ |n| p_a
p_a = 100 Δa/|a|; Δz ≈ |z| p_z/100
| Symbol | Meaning |
|---|---|
| a, b | Measured quantities. Use matching units for addition or subtraction. |
| Δa, Δb | Non-negative absolute uncertainties, each in its corresponding input unit. |
| n | An exact exponent with no uncertainty. |
| p, Δz | Percentage uncertainty and absolute output uncertainty. |
When to use this calculator
Use this method when your class or lab report asks you to add absolute uncertainties for sums, or percentage uncertainties for products and quotients. For the example, a is a voltage and b is a current, so a/b is a resistance.
These are linear worst-case estimates, not the root-sum-of-squares method for independent standard uncertainties. Products, quotients and powers use first-order approximations; they are not exact interval bounds. Correlations, repeated use of the same variable and large uncertainties need additional treatment.
Worked example
The calculator opens with these example values. All steps below are available even with JavaScript disabled.
- Input a = 4.52 ± 0.03
- pₐ = Δa / |a| × 100 = 0.03 / |4.52| × 100 = 0.6637168%
- Input b = 0.231 ± 0.002
- pᵦ = Δb / |b| × 100 = 0.002 / |0.231| × 100 = 0.8658009%
- z = (4.52) / (0.231) = 19.5671
- p_z ≈ pₐ + pᵦ = 0.6637168% + 0.8658009% = 1.529518%
- Δz ≈ |z| × p_z / 100 = |19.5671| × 1.529518 / 100 = 0.2992822
- Final rounded result: (19.6 ± 0.3) Ω
(19.6 ± 0.3) Ω
Common mistakes
- Absolute uncertainties add even when measured values are subtracted.
- Use |n| for powers: a negative exponent does not create a negative uncertainty.
- Do not use this two-measurement rule blindly for a − a or a / a, where both occurrences describe the same quantity.
- A denominator interval that includes zero cannot be summarized by a finite symmetric uncertainty.