Error Propagation Calculator (Uncertainty of Sums, Products, Quotients, Powers)
The uncertainty of A + B, A − B, A × B, A ÷ B and Aⁿ from the uncertainties of A and B, using the standard rules for independent errors.
Results
What this tool does
Every measurement carries an uncertainty, and a result computed from two of them inherits both. If the errors are independent, they do not simply add: for a sum or a difference the absolute uncertainties combine in quadrature, √(δA² + δB²), and for a product or quotient the relative uncertainties do. Adding them straight, as the worst-case line shows, is always safe but overstates the doubt. For a power Aⁿ the relative uncertainty is multiplied by |n|, so squaring a quantity doubles its percentage error. These are first-order rules, good while the uncertainties are small compared with the values.
Formula
A ± B: δ = √(δA² + δB²) · A × B: δ = √((B δA)² + (A δB)²) · A ÷ B: δ = √((δA ÷ B)² + (A δB ÷ B²)²) · Aⁿ: δ = |n| Aⁿ⁻¹ δA
Variables
| Symbol | Meaning | Unit |
|---|---|---|
va | Quantity A — measured value | — |
ua | Quantity A — uncertainty (±) | — |
vb | Quantity B — measured value | — |
ub | Quantity B — uncertainty (±) | — |
ex | Exponent n for A to the power n | — |
US | Uncertainty of A + B and of A − B (±) | — |
S | A + B | — |
D | A − B | — |
WC | Worst case if the errors line up (±) | — |
P | A × B | — |
UP | Uncertainty of A × B (±) | — |
Q | A ÷ B | — |
UQ | Uncertainty of A ÷ B (±) | — |
W | A to the power n (A must be positive) | — |
UW | Uncertainty of A to the power n (±) | — |
Worked example
- Quantity A — measured value12.5
- Quantity A — uncertainty (±)0.3
- Quantity B — measured value4.2
- Quantity B — uncertainty (±)0.1
- Exponent n for A to the power n2
- Uncertainty of A + B and of A − B (±)0.3162
- A + B16.7000
- A − B8.3000
- Worst case if the errors line up (±)0.4000
- A × B52.5000
- Uncertainty of A × B (±)1.7749
- A ÷ B2.9762
- Uncertainty of A ÷ B (±)0.1006
- A to the power n (A must be positive)156.2500
- Uncertainty of A to the power n (±)7.5000
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.