Error Propagation Calculator

Every measured number arrives with an uncertainty attached, and the moment two measurements are combined — added into a total length, multiplied into an area, divided into a density — the uncertainties combine too, by rules that are kinder than most people fear and stricter than most people practise. This calculator propagates them properly: enter two values with their uncertainties, choose the operation, and it returns the result with its combined uncertainty, using the standard root-sum-of-squares rules for independent errors. It is the arithmetic behind every honest “±” in science.

The Two Master Rules

Nearly all laboratory propagation reduces to two patterns. For addition and subtraction, work with the uncertainties themselves — the absolute uncertainties combine by root-sum-of-squares:

σ(A ± B) = √(σA² + σB²)

For multiplication and division, convert to relative (fractional or percentage) uncertainties first, combine those by the same root-sum-of-squares, then convert back:

σ(AB)/(AB) = √((σA/A)² + (σB/B)²)

Why squares and square roots, rather than simply adding? Because independent errors partially cancel: one measurement's high reading meets another's low one as often as they conspire, so the combined uncertainty grows like the diagonal of a rectangle, not the sum of its sides. Adding uncertainties linearly is the older, pessimistic convention — it remains the right treatment when the errors are correlated (the same mis-calibrated ruler measuring both lengths), because then no cancellation can be trusted. The calculator implements the independent case, which is the standard default, and names the assumption in its output.

Worked Examples: One Pair of Measurements, Four Fates

Take A = 10 ± 0.2 (2% relative) and B = 5 ± 0.1 (also 2%). Sum: 15 ± √(0.04 + 0.01) = 15 ± 0.224 — note the relative uncertainty improves to 1.5%, the reward for averaging-like operations. Difference: 5 ± 0.224 — the same ±0.224, now a painful 4.5% of the smaller result, which is why measuring a small difference of two large quantities is experimental quicksand. Product: 50, with relative uncertainty √(2%² + 2%²) = 2.83%, so 50 ± 1.41. Quotient: 2.00 ± 2.83% = 2 ± 0.057. Four operations, one pair of measurements, and the complete moral: absolute uncertainties rule sums, relative uncertainties rule products, subtraction of near-equals is where precision goes to die, and every one of those conclusions falls out of two square-root formulas this page evaluates for you.

Powers, Roots and the General Formula

The two master rules are corollaries of one master equation: for any function f of measured quantities, σf² ≈ Σ (∂f/∂xᵢ)² σxᵢ² — each input's uncertainty weighted by how steeply the output responds to it (the partial derivative), squared and summed. The famous special cases fall straight out. A power: for y = xⁿ, the relative uncertainty multiplies by |n| — a length measured to 2% gives an area (L²) to 4% and a volume (L³) to 6%, and a square root (n = ½) halves the relative uncertainty, one of the few genuine bargains in the subject. A constant multiple passes its relative uncertainty through untouched. An average of n repeated readings divides the single-reading uncertainty by √n — the same root that governs the standard error in the RSE Calculator. When a derivation matters — a thesis, a calibration certificate — write the general formula, take the derivatives, and let the two master rules check your arithmetic on the simple parts.

Reporting: How Many Digits Deserve to Live

Propagation's last discipline is typographical honesty. The convention across the sciences: quote the uncertainty to one or two significant figures, and round the value to match its decimal place — 50 ± 1.4, not 50.0000 ± 1.41421, a formatting that claims knowledge nobody has. Round the uncertainty up when in doubt (conservatism in the ± is a virtue), keep one consistent convention through a whole report, and never let the printed value carry more precision than the uncertainty supports: a density reported as 2.7036 ± 0.06 g/cm³ tells the reader the author propagated mechanically and thought never. The calculator prints four decimals so you can see the raw figure; the rounding to a reportable pair is the author's final act of judgment, and this paragraph is its instruction manual.

End to End: The Density of a Metal Sample

A complete laboratory case, run through the two master rules in sequence. A machined sample is weighed at 54.2 ± 0.2 g and its volume measured by displacement at 20.0 ± 0.5 ml. Density = 54.2 / 20.0 = 2.71 g/ml — the calculator's division case. The mass carries 0.37% relative uncertainty; the volume, read off a graduated cylinder, carries 2.5%. Root-sum-of-squares: √(0.37² + 2.5²) ≈ 2.53%, so the density is 2.71 ± 0.069, reported as 2.71 ± 0.07 g/ml — aluminium's neighbourhood, identified with a defensible error bar. Note the budget's anatomy: the volume term contributes 2.5 of the 2.53 — improving the balance would polish the wrong link entirely, while a burette or pycnometer for the volume attacks the error where it lives. That ranking of terms is the working purpose of propagation, beyond the final ±: it tells the experimenter which instrument to upgrade, and it is why the calculator prints its rule notes rather than a naked answer.

When Errors Hold Hands: The Correlated Case

One counterexample keeps the root-sum rule honest. Measure two segments of a rod with the same steel ruler, each reading ±0.1 mm for the ruler's calibration, and the calibration error is shared: if the ruler reads long, it reads long on both segments, and the total length's calibration uncertainty is the linear sum ±0.2 mm, not the root-sum ±0.141 mm. The independent reading errors (your eye, twice) still combine by root-sum. Real budgets therefore split every uncertainty into a shared (systematic) ledger and an independent (random) ledger, propagate each by its own rule, and combine the two results at the end — the architecture behind national metrology reports, and the reason this calculator prints its independence assumption in plain sight rather than burying it in a footnote.

Through a Function: The Projectile's Angle

The general formula earns one worked visit. A projectile's range rides on sin(2θ); the launch angle is set to θ = 30° with an uncertainty of ±1°. The output quantity is sin(60°) = 0.866. The derivative of sin(2θ) with respect to θ is 2cos(2θ) per radian — at 60°, 2 × 0.5 = 1.0 per radian — and 1° is 0.01745 radians, so σ ≈ 1.0 × 0.01745 ≈ 0.0175, about a 2.0% relative uncertainty in the range factor from angle-setting alone. Swap in the derivative for any function — exponentials multiply relative errors by the exponent's argument, logarithms convert relative error in x to absolute error in ln x — and the pattern of this page holds: weight each input's uncertainty by the output's sensitivity to it, square, sum, root. The calculator automates the four arithmetic operations; this derivative habit covers everything else a laboratory can build.

One Last Bargain to Remember

Close with the subject's most consoling computation: the pendulum. Its period goes as the square root of its length, T ∝ √L, so by the power rule a length measured to ±0.10% delivers a period good to ±0.05% — the root halves the relative error for free. Wherever a square root stands between your measurement and your answer — pendulum periods, resonant frequencies, standard errors themselves — nature is quietly subsidising your precision. Design measurements to route through roots when you can, and through cubes never, if you can help it.

Related Tools

More from the Statistics Calculators hub: the RSE Calculator propagates sampling error into a survey's precision, the Descriptive Statistics Calculator supplies the standard deviations that feed these formulas, and the Coefficient of Determination Calculator handles uncertainty's regression-side cousin — unexplained variation.

Frequently Asked Questions

Why not just add the uncertainties?

Linear addition assumes the errors always conspire in the same direction — true only when they are perfectly correlated. For independent errors, partial cancellation is the statistical reality, and root-sum-of-squares is its formula; linear addition overstates the combined uncertainty, sometimes by 40% or more, and can kill a viable experiment on paper.

When should I use the pessimistic linear sum?

When the uncertainties share a cause: the same instrument's calibration error across both measurements, the same standard solution, the same observer's bias. Then the errors march together and the honest combination adds them directly. Mixed cases — part shared, part independent — propagate the shared part linearly and the rest by root-sum-of-squares.

Does subtraction really keep the same uncertainty as addition?

Yes — the variance of a difference of independent quantities is the sum of the variances, identical to the variance of a sum. The values subtract; the uncertainties root-sum-square regardless. This is why small differences between large measurements (a titration's two burette readings) dominate many error budgets.

How do I propagate through a formula with three or more inputs?

Apply the general formula: square each input's uncertainty times the partial derivative of the function with respect to that input, sum, and take the root. In practice, chain this calculator pairwise — combine A and B, then combine the result with C using the matching rule — which is algebraically equivalent for the four basic operations.

What uncertainty does a counted quantity have?

Counting (radioactive decays, photons, colonies) follows Poisson statistics: a count of N carries an uncertainty of about √N, a relative uncertainty of 1/√N. Count 10,000 events and the counting uncertainty is 1% — often the floor beneath an experiment's whole error budget, set by arithmetic alone.

Percentages confuse me — absolute or relative for my case?

Sums and differences want absolute uncertainties in the units of the values; products, quotients and powers want relative ones. Convert at the boundary: relative = absolute ÷ value. The calculator keeps the two regimes labelled in its output notes so the conversion is never left implicit.