What this calculator does
When two measurements, each with its own absolute uncertainty, are added, subtracted, multiplied or divided, the uncertainty of the result is not simply the sum of the two individual uncertainties. The standard approach is to combine them in quadrature, meaning the uncertainties are squared, added and then square-rooted, which reflects that independent errors are unlikely to both fall at their maximum in the same direction at once.
This calculator works out that combined absolute uncertainty for either an add-or-subtract combination or a multiply-or-divide combination, since the two cases use a different formula: addition and subtraction combine the absolute uncertainties directly, while multiplication and division combine the relative uncertainties instead, then convert the result back to an absolute figure.
The formula
For addition or subtraction, the combined uncertainty is the square root of the sum of the squares of the two absolute uncertainties: √(ΔA² + ΔB²). For multiplication or division, each measurement's relative uncertainty (its absolute uncertainty divided by its own value) is combined the same way, and the result is multiplied back by the calculated answer to give an absolute figure again.
| Term | Meaning |
|---|---|
| A, B | The two measured values being combined. |
| ΔA, ΔB | The absolute uncertainty of each measurement, in the same units as the measurement itself. |
| Relative uncertainty | A measurement's absolute uncertainty expressed as a fraction or percentage of its own value: ΔA / A. |
| Quadrature | Combining values by squaring each, summing, then taking the square root, the standard way to combine independent uncertainties. |
The inputs explained
| Field | What to enter |
|---|---|
| Measurement A | The first measured value. |
| Absolute uncertainty of A | The absolute uncertainty on the first measurement, in the same units as A. |
| Measurement B | The second measured value. |
| Absolute uncertainty of B | The absolute uncertainty on the second measurement, in the same units as B. |
| Combining A and B by | How A and B are being combined: adding, subtracting, multiplying or dividing. |
When to use it
Adding two lab measurements
Two lengths measured with the same ruler, each with its own reading uncertainty, need their combined uncertainty worked out when the two lengths are added together end to end.
Working out a density from mass and volume
Density is mass divided by volume, and both mass and volume carry their own measurement uncertainty. Dividing them combines the two relative uncertainties into one for the density result.
Checking whether an experimental result agrees with theory
When a derived quantity is built from two or more raw measurements, the combined uncertainty tells you how wide a range the final result could plausibly fall within before comparing it against an expected value.
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 combined uncertainty change with the operation used?
The same two measurements and uncertainties, combined by each of the four operations in turn.
| Operation | Combined absolute uncertainty | Combined relative uncertainty |
|---|---|---|
| Add | 0.2236 | 1.33% |
| Subtract | 0.2236 | 2.73% |
| Multiply | 1.517 | 2.82% |
| Divide | 0.0821 | 2.82% |
Questions
What is the difference between absolute uncertainty and relative uncertainty?
Absolute uncertainty is expressed in the same units as the measurement itself, such as ± 0.2 cm. Relative uncertainty expresses that same uncertainty as a fraction or percentage of the measured value, such as 1.6%, which makes it easier to compare uncertainty across measurements of very different sizes.
Why not just add the two uncertainties together directly?
Simply adding them assumes both measurements are at their worst-case error in the same direction at the same time, which is overly pessimistic for independent measurements. Combining in quadrature (squaring, summing, then square-rooting) reflects that independent errors are more likely to partly cancel than to align.
Does the formula change for multiplying by a plain number instead of another measurement?
Yes. Multiplying a measurement by an exact constant (with no uncertainty of its own) simply scales the absolute uncertainty by that same constant, rather than needing the full quadrature formula, which is only needed when both quantities carry their own uncertainty.
Is this the same as the absolute error calculation used against a known true value?
No. Comparing a single measurement against an accepted true value gives an absolute error, a measure of accuracy. This calculator instead combines the uncertainties of two independent measurements when they are added, subtracted, multiplied or divided together, which is a measurement-propagation problem rather than an accuracy check.
To compare a single measured value against a known true value instead, see the relative and absolute error calculator.