Error Propagation Calculator

The rule behind every case

All the specific formulas come from one idea: a small change in an input produces a change in the output scaled by the partial derivative. Combining independent contributions in quadrature gives the general expression, and every rule below is just that expression evaluated for a particular function.

• Adding or subtracting combines absolute uncertainties: σf = √(σa² + σb²). Subtraction does not cancel uncertainty; it adds it.
• Multiplying or dividing combines relative uncertainties: σf/|f| = √((σa/a)² + (σb/b)²).
• Raising to a power multiplies the relative uncertainty by the exponent: σf/|f| = |n| σa/|a|. Cubing a quantity triples its percentage uncertainty.
• Multiplying by an exact constant scales the absolute uncertainty with it and leaves the relative uncertainty unchanged.

Why quadrature and not a plain sum

Adding uncertainties linearly assumes every error pushes the same way at the same time. For independent random errors that is pessimistic: they partially cancel, and the statistically correct combination is the square root of the sum of squares. Two measurements each ±0.1 combine to ±0.14, not ±0.2.

This breaks down when errors are correlated. Think of a miscalibrated instrument, a thermal drift affecting every reading, or a shared reference standard. Those are systematic, and quadrature can then be wrong in either direction: a shared offset adds in a sum but cancels in a difference. Adding the uncertainties linearly gives a safe upper bound.

Rounding, and why it matters

Quote the uncertainty to one significant figure, or two when the leading digit is 1, then round the value to the same decimal place. So 4.28 ± 0.05, never 4.28371 ± 0.05. The extra digits claim a precision the uncertainty explicitly denies. This calculator applies that convention automatically and also shows the unrounded value if you need to carry it into a further calculation.

Carry full precision through intermediate steps and round only at the end. Rounding early injects extra error that then propagates. For a number with no stated uncertainty, the significant figures calculator counts its figures, rounds it and carries the figures through a calculation instead.

Where the error budget usually sits

Because contributions add in quadrature, the largest one dominates quickly. If one term is three times another, it accounts for 90% of their combined variance, and refining the smaller terms is wasted effort. Compute the budget before you decide what to improve. The answer is often that a single measurement is worth doing better and the rest are already good enough.