What this calculator does
When two measured quantities are combined, their uncertainties combine too, but not by simple addition. Uncertainties add in quadrature, meaning the square root of the sum of squares, because independent errors partly cancel rather than always reinforcing.
Which quantities add depends on the operation. For sums and differences the absolute uncertainties combine; for products and quotients the relative uncertainties do. That distinction is the whole subject in one sentence, and getting it the wrong way round is the most common mistake in the topic.
The formula
For a sum or difference, the absolute uncertainty is the square root of the sum of the squared absolute uncertainties. For a product or quotient, the relative uncertainty is the square root of the sum of the squared relative uncertainties, which is then multiplied by the result to give the absolute figure. Both formulas assume the two uncertainties are independent.
| Term | Meaning |
|---|---|
| Absolute uncertainty | The uncertainty in the original units, such as ±0.5 cm. |
| Relative uncertainty | The uncertainty as a fraction or percentage of the value. |
| In quadrature | Combining as the square root of a sum of squares, which is how independent errors combine. |
| Independence | The assumption that the two errors are unrelated. Correlated errors combine differently and usually more severely. |
The inputs explained
| Field | What to enter |
|---|---|
| X | The first measured value. |
| Uncertainty ΔX | Its absolute uncertainty. |
| Y | The second measured value. |
| Uncertainty ΔY | Its absolute uncertainty. |
| Operation | How the two are combined. Sums and differences use absolute uncertainties; products and quotients use relative ones. |
When to use it
Reporting a laboratory result
Any calculated quantity derived from measurements needs its uncertainty propagated to be reported honestly.
Deciding which measurement to improve
Because errors add in quadrature, the largest one dominates, and improving a small one barely helps.
Checking whether a difference is meaningful
Subtracting two similar measurements inflates the relative uncertainty sharply, which is worth seeing before drawing a conclusion.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the operation change the uncertainty?
The same two measurements combined four ways.
| Operation | Result Z | Absolute uncertainty ΔZ | Relative uncertainty |
|---|---|---|---|
| X + Y | 17.000 | 0.5385 | 3.17% |
| X − Y | 7.000 | 0.5385 | 7.69% |
| X × Y | 60.000 | 3.466 | 5.78% |
| X ÷ Y | 2.400 | 0.1386 | 5.78% |
Questions
Why do uncertainties add in quadrature rather than directly?
Because independent errors are as likely to partly cancel as to reinforce. Adding them directly assumes the worst case happens every time, which systematically overstates the uncertainty. The square root of the sum of squares is the statistically correct combination for independent errors.
Why does subtraction make the relative uncertainty worse?
Because the absolute uncertainty stays the same while the result gets smaller. Subtracting two close measurements is the classic trap: 12.0 ± 0.5 minus 11.5 ± 0.5 gives 0.5 ± 0.71, an uncertainty larger than the answer itself.
Which measurement should I improve first?
The one with the largest uncertainty, by a wide margin. Because the contributions are squared, an uncertainty half the size of another contributes only a quarter as much. Refining an already-small uncertainty is usually wasted effort.
What if the errors are not independent?
Then quadrature understates the combined uncertainty and a covariance term is needed. Correlated errors commonly arise from a shared instrument, a shared calibration or a shared operator, so measuring two quantities the same way is precisely when this assumption is at risk.
For a single measurement uncertainty, see the absolute uncertainty calculator. For expressing error as a percentage, see the relative error calculator.