fundamentals · 5 min read

Error propagation basics

A result inherits the uncertainty of its inputs, and the way those uncertainties combine depends on the operation. A sum adds absolute uncertainties in quadrature, a product adds relative ones, and a power scales the relative uncertainty by the exponent, so using the wrong rule is an easy way to overstate or understate precision.

This guide covers the rules behind the error propagation calculator, the independence assumption that makes quadrature valid, the rounding convention used to report the result, and worked examples for the main modes.

By FormatFlow EditorialUpdated October 21, 2026More Scientific Tools →

Absolute and relative uncertainty are the same information

An absolute uncertainty carries the measurement's units: 2.0 ± 0.1 g means the spread is 0.1 g. The relative uncertainty divides by the value, so the same measurement has a relative uncertainty of 0.05, or 5%. Neither is more honest than the other; the operation decides which one is convenient.

Sums and differences are naturally stated in absolute terms, because adding two lengths does not change their units, while products and quotients mix quantities and are easier to reason about as fractions. Converting between the two is simple: δx/|x| is the relative uncertainty, and multiplying it by |x| returns the absolute one.

The four propagation rules

For a sum or difference z = a ± b ± …, the absolute uncertainties combine in quadrature: δz = √(δa² + δb² + …). The signs of the terms do not matter, because a negative term is just as uncertain in magnitude as a positive one.

For a product or quotient z = a × b ÷ …, the relative uncertainties combine in quadrature: δz/|z| = √((δa/|a|)² + (δb/|b|)² + …). Multiplying by |z| converts the result back to an absolute uncertainty.

For a power z = aⁿ, the relative uncertainty is scaled by the exponent: δz/|z| = |n| × δa/|a|. For an arbitrary smooth function z = f(x, y, …), the partial derivatives enter the same quadrature: δz = √(Σ(∂f/∂xᵢ × δxᵢ)²).

Sum or difference: absolute uncertainties in quadrature.
Product or quotient: relative uncertainties in quadrature.
Power: relative uncertainty multiplied by |n|.
Custom function: partial derivatives times each input uncertainty, in quadrature.

Why quadrature, and when it fails

Quadrature assumes the input errors are independent, random and roughly symmetric, so their variances add. Independent random errors can partially cancel, and combining them with the square root of the sum of squares reflects that; simply adding 0.1 and 0.2 would give 0.3, while quadrature gives about 0.22, because the two errors rarely peak in the same direction at once.

The assumption fails for correlated or systematic errors. Two masses weighed on the same badly calibrated balance share a bias that does not average out, and combining their uncertainties in quadrature would understate the error in a sum. Correlated inputs need a covariance term, which the calculator does not model, so state the independence assumption whenever you quote a combined uncertainty.

Worked product example

Take a quantity z = a × b with a = 2.0 ± 0.1 and b = 3.0 ± 0.2. The relative uncertainties are 0.05 and about 0.0667, so δz/|z| = √(0.05² + 0.0667²) ≈ 0.0833, or about 8.3%. The central value is 2.0 × 3.0 = 6.0, so the absolute uncertainty is 6.0 × 0.0833 ≈ 0.5 and the unrounded result is 6.0 ± 0.5.

Rounding follows the reporting convention: the uncertainty 0.5 has one significant figure, and the central value is rounded to the same decimal place, so the reported form is 6.0 ± 0.5 with a relative uncertainty of about 8.3%. The same inputs combined with the sum rule instead would give a much smaller absolute uncertainty, which is why the operation, not the data, chooses the rule.

Reporting a result that can be checked

Round the uncertainty to one significant figure, or two when the leading digit is 1, and round the central value to the same decimal place. Keep the unrounded values for any further calculation, and only round the form you report; the uncertainty, not the other digits, decides where the value stops.

Give the units once, state whether the uncertainty is a standard deviation, standard error or tolerance, and say which propagation rule and independence assumption were used. A reader who can repeat the arithmetic and see the assumptions cannot be misled by a precise-looking result.

Sums and differences: absolute uncertainties in quadrature.
Products and quotients: relative uncertainties in quadrature.
Powers: relative uncertainty times the absolute exponent.
Round the uncertainty first, then the value to the same decimal place.
Name the uncertainty type and the independence assumption.

Frequently asked questions

When should I combine absolute uncertainties and when relative ones?

Use absolute uncertainties for sums and differences, because the result has the same units as the terms. Use relative uncertainties for products, quotients and powers, then convert back to an absolute uncertainty at the end. Converting between the two is just dividing or multiplying by the value.

Why not simply add the uncertainties together?

Adding them assumes the errors always point the same way at the same time, which overstates the combined uncertainty for independent random errors. Quadrature assumes the errors partly cancel, and it reduces to simple addition only for perfectly correlated errors, which need a covariance term instead.

What if the input errors are correlated or systematic?

Quadrature is not valid then. Errors sharing a calibration bias or a common drift do not cancel, and the combined uncertainty needs covariance terms or a direct propagation of the bias. Report the limitation rather than applying the independent-error formulas.

How should I round a value and its uncertainty?

Round the uncertainty to one significant figure, or two when its leading digit is 1, then round the central value to the same decimal place and keep the units. The convention avoids implying more precision than the measurement supports and keeps the reported form consistent.

Can I propagate uncertainty through an arbitrary formula?

Yes, for a smooth function: take the partial derivative with respect to each variable, multiply by that variable's uncertainty, and combine the terms in quadrature. The calculator estimates the derivatives numerically, so very steep or non-smooth functions deserve a hand check.

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