Error Propagation Calculator: Sums, Differences, Products and Powers

Give two measured quantities with their uncertainties and the combination you need, and this page returns the uncertainty on the answer, where to cut the figures, and both rules side by side — the simple addition your board marks, and the quadrature of the metrology guide, which on these defaults is 2.13 per cent against 3.00.

Last updated 11 Nov 2026

Uncertainty in a derived quantity

A ± ΔA and B ± ΔB, combined

A — first measured quantityAny quantity in any unit. The rules below are all about ratios, so a diameter in millimetres and a mass in grams propagate identically.

ΔA — its absolute uncertaintyIn the same unit as A, and positive. One least count, one half-division, or the spread of repeats. The page converts it to a percentage for you, because percentages are what combine.

B — second measured quantityFor a single-quantity job such as the volume of a sphere from its radius, put the radius in A with its uncertainty, set B to 1 with zero uncertainty, and give B the power 0.

ΔB — its absolute uncertaintySame unit as B. A quantity known exactly — a count, a defined constant, the 2 in 2πr — gets zero here and then contributes nothing, which is the correct behaviour and not an approximation.

The combinationSums and differences combine ABSOLUTE uncertainties; products, quotients and powers combine PERCENTAGE uncertainties. That is the whole of the subject, and mixing the two is the commonest mistake on it. The third option covers everything multiplicative: n = m = 1 is a product, m = −1 a quotient, n = 2 a square, n = 0.5 a square root.

n — the power on AUsed only in the third combination. A power multiplies the percentage uncertainty by itself: the volume of a sphere carries three times the percentage uncertainty of its radius, and the period of a pendulum carries half that of its length.

m — the power on BNegative for a quotient: density from m and V is m¹V⁻¹, so m = −1. The SIGN of a power never matters to the uncertainty — dividing is exactly as lossy as multiplying — which is why the page takes the modulus of both powers.

3.002%Example

A rectangle measured as A = 12.4 ± 0.2 cm by B = 3.60 ± 0.05 cm; its area, by the rule a board will mark

Absolute for sums, percentage for products, and the power multiplies

R = A ± B: ΔR = ΔA + ΔB · R = AⁿB^m: ΔR/R = |n| ΔA/A + |m| ΔB/B · in quadrature, each sum becomes a root of squares

ΔA + ΔB: the sum and difference rule, in ABSOLUTE units. It is the same for both signs: a difference does not subtract the uncertainties, it adds them. This is the half everyone gets right
|n| ΔA/A: the product, quotient and power rule, in RELATIVE terms. The modulus is deliberate: a quotient loses exactly as much as a product, because the sign of the power never reaches the uncertainty
the power n: a straight multiplier on that quantity’s percentage uncertainty. One per cent on a radius is two on the area and three on the volume; a square root halves it, which is why a pendulum’s period is more forgiving of a length error than you would guess
the quadrature form: ΔR = √(ΔA² + ΔB²) and its relative twin. JCGM 100:2008 calls this the combined standard uncertainty and its clause 5.1.2 names equations (10) and (13) “the law of propagation of uncertainty”. It assumes the two errors are independent and is always the smaller answer
where to cut: round the uncertainty to one significant figure, then round R to that same decimal place. Here 44.64 with ±1.34 becomes 45 ± 1, and writing 44.64 ± 1.34 claims four figures that the uncertainty denies

Worked example

A rectangle measured as A = 12.4 ± 0.2 cm by B = 3.60 ± 0.05 cm; its area, by the rule a board will mark
TURN BOTH UNCERTAINTIES INTO PERCENTAGES, because an area is a product and products combine relative uncertainties. ΔA/A = 0.2/12.4 = 1.613 per cent; ΔB/B = 0.05/3.60 = 1.389 per cent. Note what has happened: the larger absolute uncertainty is on A, but the two percentages are nearly equal, and it is the percentages that matter.
ADD THEM, WHICH IS WHAT THE SYLLABUS ASKS FOR. 1.613 + 1.389 = 3.002 per cent. Cambridge 9702 puts it plainly: “assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties”. Each power is 1, so neither percentage is multiplied up.
CONVERT BACK TO AN ABSOLUTE UNCERTAINTY. R = 12.4 × 3.60 = 44.64 cm², and 3.002 per cent of that is ±1.34 cm². The quadrature figure for comparison is √(1.613² + 1.389²) = 2.128 per cent, or ±0.95 cm² — 1.41 times smaller, which is the usual factor when two contributors are about equal.
ROUND THE UNCERTAINTY FIRST, THEN THE VALUE. One significant figure on 1.34 gives ±1, which sits in the units column, so R rounds to the units column too: (45 ± 1) cm². Writing 44.64 ± 1.34 claims four figures in a quantity whose second is already in doubt.
AND CHECK WHICH MEASUREMENT TO IMPROVE. A contributes 53.7 per cent of the total and B contributes 46.3, so neither dominates and improving either alone buys little — halving A's uncertainty takes the total from 3.00 to 2.20 per cent. Had the rectangle been a cube of side A instead, the power of 3 would have made A's share overwhelming, and that is where the effort would belong.

The five combinations, with both rules side by side

The A − B row and the A / B row are the two that catch people. A difference adds the absolute uncertainties exactly as a sum does — it never subtracts them — and then divides by a smaller number, so the percentage rises for free. And a quotient is governed by exactly the same expression as a product: the sign of the power never reaches the uncertainty, so halving is as lossy as doubling. The last row is the one to remember in the laboratory: a power multiplies the percentage uncertainty before anything else happens, so in any experiment with a cube in it, the cubed quantity is where the measuring time should go.
R is	Combine	Added rule (CBSE, CIE)	Quadrature (JCGM 100:2008)	Worked on A = 12.4 ± 0.2, B = 3.60 ± 0.05
A + B	absolute	ΔA + ΔB	√(ΔA² + ΔB²)	16.00 ± 0.25 (1.56 %); quadrature ±0.21
A − B	absolute	ΔA + ΔB — still added	√(ΔA² + ΔB²)	8.80 ± 0.25 (2.84 %); the percentage rose because R shrank
A × B	relative	ΔA/A + ΔB/B	√((ΔA/A)² + (ΔB/B)²)	44.64 ± 1.34 (3.00 %); quadrature 2.13 %
A / B	relative	ΔA/A + ΔB/B — identical to the product	same as the product	3.444 ± 0.103 (3.00 %); dividing is no safer than multiplying
Aⁿ	relative	\|n\| × ΔA/A	\|n\| × ΔA/A	A³ gives 4.84 % from A’s 1.61 %; √A gives 0.81 %

Where the two rules land, as the contributors change

Two things fall out of this table. The gap between the rules is widest when the contributors are equal, where the added rule is always √2 times the quadrature one, and it narrows to nothing as one contributor takes over — which is why a dominated uncertainty budget hardly cares which rule you use. And the second column shows how little a small contributor buys: moving from 5.0 and 1.0 to 5.0 and 0.5 improves the quadrature total from 5.10 to 5.02 per cent. That is the arithmetic behind the advice to measure the dominant quantity better and leave the others alone.
Two percentage uncertainties	Added	Quadrature	Added ÷ quadrature
1.0 % and 1.0 %	2.00 %	1.41 %	1.41
1.613 % and 1.389 % (this page’s defaults)	3.00 %	2.13 %	1.41
2.0 % and 1.0 %	3.00 %	2.24 %	1.34
3.0 % and 1.0 %	4.00 %	3.16 %	1.26
5.0 % and 1.0 %	6.00 %	5.10 %	1.18
5.0 % and 0.5 %	5.50 %	5.02 %	1.10

Absolute uncertainties for sums, percentages for products, and the power that multiplies both

There are only two rules, and the subject is knowing which one the operation calls for. For a sum or a difference, the ABSOLUTE uncertainties add. For a product, quotient or power, the PERCENTAGE uncertainties add, each multiplied by the magnitude of its power first. Mixing them is the commonest mistake here, and it is detectable by a glance at the units: an absolute uncertainty carries the unit of the quantity and a percentage carries none, so an expression adding one to the other is wrong before any arithmetic happens.

Two consequences surprise people. The first is that a difference ADDS uncertainties, never subtracts them: the two errors might lie either way, so the worst case has them opposed. Then, because the difference is small, the percentage rises for nothing — 3.8 minus 3.6, each good to 0.05, is 0.2 ± 0.1, fifty per cent out of two measurements each good to one and a half. The second is that a quotient is governed by exactly the same expression as a product, because the sign of a power never reaches the uncertainty. Dividing is precisely as lossy as multiplying, and the modulus in the formula is doing real work.

The power is where good measurements produce poor answers. A percentage uncertainty is multiplied by the power before it is combined with anything, so a radius known to one per cent gives an area known to two and a volume known to three; a square root halves it, which is why a pendulum’s period is forgiving of an error in its length. The laboratory consequence is direct. The quantity to measure more carefully is not the one with the largest uncertainty but the one with the largest uncertainty times power — and the contribution rows above compute exactly that, so the decision need not be guessed.

And there are two rules for combining, not one, which is the thing nothing else on the web arbitrates. Cambridge 9702 asks candidates to “assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties”, and the word quadrature does not appear anywhere in that syllabus. Addition is a worst case: it assumes both measurements erred the same way. JCGM 100:2008, which every calibration laboratory works to, takes the square root of the sum of squares instead — its clause 5.1.2 names that “the law of propagation of uncertainty” — because independent errors rarely conspire. With two equal contributors the added rule is √2 larger, every time. Neither is wrong, so this page leads with the one a school answer sheet wants and prints the other beside it.

Frequently asked questions

Do I add or subtract the uncertainties when I subtract two quantities?

You add them. The uncertainties of a difference combine exactly as those of a sum: ΔR = ΔA + ΔB. The reason is that the two errors could lie in opposite directions, so the gap between the two values can be wrong by the sum of both. Subtracting uncertainties would claim that measuring two things badly gives a difference known perfectly, which is plainly false. The sting is in the percentage: the absolute uncertainty is unchanged while the answer has shrunk, so a difference of nearly equal quantities has a large percentage uncertainty however good the two measurements were.

Should I add percentage uncertainties or combine them in quadrature?

For a school paper, add them. Cambridge 9702 asks for “simple addition of absolute or percentage uncertainties” and never mentions quadrature; CBSE’s Unit I speaks of “Determining the uncertainty in result” and the linear rule is what its textbooks and examiners use. Quadrature — the square root of the sum of squares — is the combined standard uncertainty of JCGM 100:2008 and is what a calibration laboratory computes, because it assumes the errors are independent rather than conspiring. It is always the smaller figure, by exactly √2 when two contributors are equal. Quote the added value as your answer and the quadrature value as a comment if you want to show you know both.

Why does a power multiply the percentage uncertainty?

Because a small fractional change in a quantity produces n times that fractional change in its nth power. Make a radius one per cent too large and the area is about two per cent too large and the volume about three. So a radius known to one per cent gives a sphere’s volume known to three, and the practical rule follows: in an experiment where something is cubed, that measurement deserves the most care, whatever its own uncertainty looks like beside the others. A fractional power works the same way downwards — a square root halves the percentage uncertainty, which is why a pendulum’s period tolerates a length error better than you would guess.

What if one of my quantities is known exactly?

Give it zero uncertainty and it drops out, which is the correct behaviour rather than an approximation. A count of oscillations, the 2 in 2πr, π itself and any conversion factor fixed by definition in the SI contribute nothing to the uncertainty however large they are, because they have no fractional error to contribute. The mistake worth avoiding is the opposite one: treating a quantity as exact because it was printed rather than measured. A value marked on a component is a nominal figure with a tolerance, and the tolerance belongs in the budget.

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References

Cambridge Assessment International Education, Cambridge International AS & A Level Physics 9702 syllabus for examination in 2025, 2026 and 2027, version 1 (cambridgeinternational.org, read 11 November 2026). Section 1.3 asks candidates to “understand and explain the effects of systematic errors (including zero errors) and random errors in measurements”, to “understand the distinction between precision and accuracy” and to “assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties”. On figures: “If this quantity has n significant figures, then n or n+1 significant figures are appropriate for the calculated value”. Read in full because of what is absent: the word quadrature never appears.
JCGM 100:2008(E), Evaluation of measurement data — Guide to the expression of uncertainty in measurement, BIPM for the Joint Committee for Guides in Metrology, first edition September 2008, corrected 2010 (read 11 November 2026). The source of the quadrature rule. Clause 5.1.2 defines the combined standard uncertainty as “the positive square root of the combined variance” given by its equation (10), and states that equations (10) and (13) “express what is termed in this Guide the law of propagation of uncertainty”. A metrology guide, not a syllabus: cited to say where the rule comes from, not to suggest a board will mark it.
CBSE, Senior School Curriculum 2025-26: Physics (Code 042), Classes XI-XII (cbseacademic.nic.in, read 11 November 2026). Unit I lists “significant figures, Determining the uncertainty in result” and “Dimensions of physical quantities, dimensional analysis and its applications”. The Class XI practical list is where the instrument pages come from: Section A experiment 1 measures “diameter of a small spherical/cylindrical body” and the “internal diameter and depth of a given beaker/calorimeter using Vernier Callipers”, experiment 2 the “diameter of a given wire and thickness of a given sheet using screw gauge”, and Activity 1 is “To make a paper scale of given least count, e.g., 0.2cm, 0.5 cm”. Cited, not endorsed.
NCERT, Physics: Textbook for Class XI, Part I, Chapter 1, “Units and Measurements” (keph101.pdf, read from ncert.nic.in on 11 November 2026). The source of every significant-figure and rounding rule here, quoted not paraphrased: “All the non-zero digits are significant”; “All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all”; “The terminal or trailing zero(s) in a number without a decimal point are not significant”; “The trailing zero(s) in a number with a decimal point are significant”. Also 1.3.1 (×/÷ keeps the fewest significant figures, +/− the fewest decimal places), 1.3.2 (at a dropped five, “if the preceding digit is even, the insignificant digit is simply dropped” and “if it is odd, the preceding digit is raised by 1”), 1.5 dimensional formulae and 1.6.1 homogeneity. Worth recording what this rationalised edition does NOT contain: no section on errors in measurement and none on combination of errors — that phrase appears once inside an example in 1.3.3, and vernier callipers and the screw gauge appear only in the exercises.