Error Propagation Calculator
Enter two measured values and their absolute uncertainties, choose an operation, and this calculator applies the standard quadrature (Gaussian) error propagation rule to give you the result and its propagated uncertainty. It covers addition, subtraction, multiplication, division, power-law, and natural logarithm, with a step-by-step panel showing every intermediate calculation.
What is error propagation?
Every physical measurement carries some uncertainty, whether from instrument precision, environmental variation, or reading error. When you combine two or more uncertain quantities through a mathematical operation, those individual uncertainties combine into an uncertainty in the final result. This process is called error propagation (also known as propagation of uncertainty). Knowing the final uncertainty is essential in any quantitative science: without it you cannot assess whether two measurements agree, whether a result is significant, or how much you should trust a derived quantity.
Quadrature rule vs. linear (worst-case) rule
The most widely used method is the quadrature (Gaussian or root-sum-of-squares) rule. It assumes the individual uncertainties are independent and random, so they partially cancel on average. The propagated uncertainty is the square root of the sum of squared contributions. The linear (worst-case) rule instead adds the absolute contributions directly, giving a conservative upper bound that applies when errors are fully correlated or when you want a safe maximum. In practice, the quadrature rule is almost always appropriate for experimental measurements with independent sources of error, while the linear rule suits systematic errors that always act in the same direction.
How each operation propagates uncertainty
For addition and subtraction, the absolute uncertainties combine: dZ = sqrt(dX^2 + dY^2). For multiplication and division, the relative (fractional) uncertainties combine: dZ/|Z| = sqrt((dX/X)^2 + (dY/Y)^2), so the absolute uncertainty is that fraction of |Z|. For a power Z = X^n, the chain rule gives dZ = |n| * |X|^(n-1) * dX, meaning high exponents amplify the relative uncertainty of X by a factor of |n|. For the natural logarithm Z = ln(X), the formula simplifies to dZ = dX / |X|, because the derivative of ln(X) is 1/X. All of these are special cases of the general formula: dZ = sqrt(sum of (partial dZ / partial Xi * dXi)^2) for independent variables.
Practical tips for reducing uncertainty
Identify which input contributes the most to the final uncertainty (the dominant error source) and focus precision improvements there. For multiplied quantities, both relative uncertainties matter equally, so improving the less precise measurement first gives the best return. For a power Z = X^n with large |n|, the relative uncertainty of X is magnified by |n|, making X the critical measurement. Use the step-by-step panel and the chart of dZ vs. dX to see how the total propagated uncertainty responds to changes in each input uncertainty.
Error propagation formulas for common operations
| Operation | Formula for Z | Propagated uncertainty dZ (quadrature) | Propagated uncertainty dZ (linear / worst-case) |
|---|---|---|---|
| Addition | Z = X + Y | sqrt(dX^2 + dY^2) | dX + dY |
| Subtraction | Z = X - Y | sqrt(dX^2 + dY^2) | dX + dY |
| Multiplication | Z = X * Y | |Z| * sqrt((dX/X)^2 + (dY/Y)^2) | |Z| * (dX/|X| + dY/|Y|) |
| Division | Z = X / Y | |Z| * sqrt((dX/X)^2 + (dY/Y)^2) | |Z| * (dX/|X| + dY/|Y|) |
| Power | Z = X^n | |n| * |X|^(n-1) * dX | |n| * |X|^(n-1) * dX |
| Natural log | Z = ln(X) | dX / |X| | dX / |X| |
Quadrature rule (independent errors). dX and dY are absolute uncertainties; X, Y are measured values; Z is the result.
Frequently asked questions
What is the difference between absolute and relative uncertainty?
Absolute uncertainty (dZ) has the same units as the measured quantity and states the size of the error band directly, for example 5.0 +- 0.1 metres. Relative (or fractional) uncertainty is the ratio dZ / |Z|, often expressed as a percentage, and describes the error as a proportion of the result. Relative uncertainty is useful for comparing the precision of measurements with different magnitudes or units.
When should I use the quadrature rule vs. the worst-case rule?
Use the quadrature rule (root-sum-of-squares) when your uncertainties come from independent, random sources, which is the standard assumption for instrument reading errors and random fluctuations. Use the linear (worst-case) rule when errors are correlated or systematic, for example when both X and Y are measured with the same uncalibrated instrument whose zero offset affects both values the same way. The worst-case rule gives a larger, more conservative bound and is rarely necessary in routine experimental work.
Why does subtraction have the same uncertainty formula as addition?
Because the propagated uncertainty depends on how sensitive Z is to small changes in X and Y, which is the partial derivative. For Z = X - Y, the partial derivative with respect to Y is -1, just as for addition it is +1. When you square these to add them in quadrature, the sign disappears, so both addition and subtraction give dZ = sqrt(dX^2 + dY^2). In practical terms, subtracting two similar quantities does not reduce uncertainty - it keeps it the same - while the result Z gets much smaller, so the relative uncertainty can become very large.
What happens to relative uncertainty under multiplication and division?
For Z = X * Y or Z = X / Y, the relative uncertainty of Z equals the quadrature combination of the relative uncertainties of X and Y: dZ/|Z| = sqrt((dX/X)^2 + (dY/Y)^2). This means that if dX/X = 2% and dY/Y = 3%, the relative uncertainty of Z is sqrt(4 + 9)% = sqrt(13)% which is about 3.6%. The measurement with the larger relative uncertainty dominates, and improving the more imprecise measurement first is the most efficient strategy.
How does raising to a power affect uncertainty?
For Z = X^n, the relative uncertainty of Z is |n| times the relative uncertainty of X: dZ/|Z| = |n| * dX/|X|. Squaring (n=2) doubles the relative uncertainty; cubing (n=3) triples it. This is why measuring radius precisely is critical when computing volume (which scales as radius^3). The formula dZ = |n| * |X|^(n-1) * dX follows from applying the chain rule (partial derivative) to X^n.
Can I propagate uncertainty through more than two variables?
Yes. The general quadrature formula extends to any number of independent variables: dZ = sqrt(sum of (partial dZ/partial Xi)^2 * dXi^2). For a chain of operations you can apply this calculator step by step, using the dZ from one step as the input dX for the next. For complex formulas with many variables, dedicated symbolic propagation tools (such as the julianibus.de or errorpropagation.com calculators) can handle arbitrary expressions automatically.