Combining measurement uncertainty: when root-sum-of-squares applies
Combine independent standard uncertainties, compare variance contributions, and avoid treating the result as a guaranteed error limit.
Method
A combined standard uncertainty describes the spread associated with a measurement result under a stated model. For independent contributions already expressed in the result’s unit, square each standard uncertainty, add the squares and take the square root. This root-sum-of-squares method does not simply add error limits. Its assumptions matter as much as the arithmetic.
First put every contribution on a standard-uncertainty basis. A specification limit, an expanded uncertainty and a standard deviation are not interchangeable labels. If a source reports expanded uncertainty with a known coverage factor, divide by that factor before combining. Apply the measurement model’s sensitivity coefficient to each component so that every entered contribution is in the output unit. This tool assumes those conversions have already been made.
Illustrative values, using the method described above.
View data
- Squared components
- 3² + 4² + 0²
- Sum of variances
- 9 + 16 + 0 = 25
- Combined standard uncertainty
- √25 = 5
Worked example
Suppose two independent contributions are 3 and 4 in the same output unit, with a third unused contribution set to zero. Their squared contributions are 9 and 16. The sum is 25 and the combined standard uncertainty is 5, rather than 7. Enter 3, 4 and 0 in the calculator to reproduce this example. No particular physical unit is implied.
The two components account for 36% and 64% of the variance total. These are shares of squared uncertainty, not shares of the measured value. If the second contribution is reduced from 4 to 2 while the first stays 3, the combined value becomes √13, about 3.606. This comparison can help identify where improving a measurement process has the largest effect under the same model.
u² = 3² + 4² + 0² = 25
u = √25 = 5
9 ÷ 25 = 36%; 16 ÷ 25 = 64%
Illustrative values, using the method described above.
View data
- First component
- 9 Output unit squared 3² = 9; 36%
- Second component
- 16 Output unit squared 4² = 16; 64%
Checks and limits
Correlated components require covariance terms and must not be combined with this independent-only shortcut. A zero component means no contribution in this model, not missing evidence that may be ignored. The result is not an expanded uncertainty, confidence statement or guaranteed maximum error. Report the measurement model, component sources, units and any coverage factor used; keep extra precision during calculation and choose final reporting precision deliberately.