Calculation
uc = √(u1² + u2²)
When adding or subtracting independent measured quantities, it is the squares of the uncertainties that add, not the uncertainties themselves.
Dubbel Meßtechnik W 1.4 Gaußsches Fehlerfortpflanzungsgesetz, Summen-/Differenzfunktion: Δy = √(ΔA² + ΔB²); GUM/JCGM 100
Uncertainty propagation when adding two measured quantities
When adding or subtracting independent measured quantities, it is the squares of the uncertainties that add, not the uncertainties themselves.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Combined standard uncertainty uc — Resulting standard uncertainty of the sum or difference of both quantities.
- Standard uncertainty of the first quantity u1 — Uncertainty of the first individual measurement, in the same unit as u2.
- Standard uncertainty of the second quantity u2 — Uncertainty of the second individual measurement, in the same unit as u1.
Example
u1=0.5 and u2=0.3 (same unit) give uc=√(0.25+0.09)≈0.583.
Assumptions and limits
Both quantities independent (uncorrelated) and normally distributed; correlated quantities, more than two terms and other combinations (product, quotient) require a different propagation formula.
Understand Uncertainty propagation when adding two measured quantities
This calculator determines the combined standard uncertainty of a sum or difference of two independent measured quantities, following classical uncertainty propagation rules.
What does this quantity describe?
For a linear combination y=x1±x2 of two independent quantities, it is not the individual uncertainties that add but their squares: uc=√(u1²+u2²) (Gaussian propagation for uncorrelated quantities).
Formula and variables
uc = √(u1² + u2²)
uc = √(u1² + u2²)
| Symbol / input | Meaning |
|---|---|
| Combined standard uncertainty uc | Resulting standard uncertainty of the sum or difference of both quantities. |
| Standard uncertainty of the first quantity u1 | Uncertainty of the first individual measurement, in the same unit as u2. |
| Standard uncertainty of the second quantity u2 | Uncertainty of the second individual measurement, in the same unit as u1. |
Choose the inputs correctly
u1 and u2 are the standard uncertainties of the two independent individual measurements, in the same unit.
How to use the calculator
Take u1 and u2 from calibration certificates, repeated measurements, or manufacturer accuracy specifications.
Worked example
u1=0.5 and u2=0.3 (same unit) give uc=√(0.25+0.09)≈0.583.
Understand the result and units
Combined uncertainty is always smaller than the plain sum u1+u2 but larger than either individual uncertainty; the larger individual uncertainty dominates the contribution.
u1, u2 and uc are quantities in the same unit as the underlying measurements; shown here as dimensionless.
Typical applications
Stating measurement uncertainty for computed sum or difference quantities, e.g. a length from two partial measurements or a force from two sensor signals.
Assumptions, limits and common mistakes
Only for two independent (uncorrelated), normally distributed quantities and a purely additive or subtractive combination; correlated quantities, more than two terms, and multiplicative combinations require different formulas.
Common mistake: Do not add the standard uncertainties arithmetically; this would systematically overestimate the combined uncertainty.
Frequently asked questions
What is “Uncertainty propagation when adding two measured quantities” used for?
Stating measurement uncertainty for computed sum or difference quantities, e.g. a length from two partial measurements or a force from two sensor signals.
Where do the input values come from?
u1 and u2 are the standard uncertainties of the two independent individual measurements, in the same unit.
What does the result not cover?
Only for two independent (uncorrelated), normally distributed quantities and a purely additive or subtractive combination; correlated quantities, more than two terms, and multiplicative combinations require different formulas.