Uncertainty & measurement
Uncertainty propagation
Combine uncertainties using school-lab rules.
Your measurements
Number formats & units
Use decimal dots in datasets. Single values also accept decimal commas and scientific notation such as 1.2e-3 or 1.2 × 10⁻³.
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Result & working
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Interpretation & assumptions
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Formula & example
The relationship
What the variables mean
| Symbol | Meaning |
|---|---|
| Measured quantities. Use matching units for addition or subtraction. | |
| Non-negative absolute uncertainties, each in its corresponding input unit. | |
| An exact exponent with no uncertainty. | |
| Percentage uncertainty and absolute output uncertainty. |
Worked example
- Operation
- Division: a / b
- Value a
- 4.52
- Absolute uncertainty in a
- 0.03
- Value b
- 0.231
- Absolute uncertainty in b
- 0.002
- Result unit (optional)
- Ω
- Uncertainty significant figures
- 1 significant figure
(19.6 ± 0.3) Ω
Follow the example steps
- 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
Common mistakes
Absolute uncertainties add for a difference.
Incorrect approach
(10 ± 0.2) − (4 ± 0.1): Δz = 0.2 − 0.1
Correct approach
z = 6; Δz = 0.2 + 0.1 = 0.3
More checks before you submit
- 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.
Method notes
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.