Error Propagation Calculator
Carry measurement uncertainties through any formula. Type the formula and each value with its ± uncertainty; get the result rounded the way a lab report wants it, the partial derivative for every input, which measurement dominates the error, and a Monte Carlo check.
How to Use
- Type the formula, with the result’s name on the left if you like: g = 4*pi^2*L/T^2. Use * between letters, ^ for powers, and sqrt(), sin(), ln() or exp() for functions.
- List each measured value on its own line as name = value ± uncertainty, for example T = 2.007 ± 0.005 s. Write a percentage as R = 100 ± 5% and concise notation as T = 2.007(5).
- Add the result’s unit if you want it printed with the answer, and choose how many significant figures the uncertainty keeps.
- Pick a mode: Partial derivatives for the standard result, Simple rules to see the sum and product rules at work, or Monte Carlo to check the answer with 10,000 random draws.
- Read the rounded result and the bar chart of which measurement contributes most of the error, then open Show Work for every derivative and term.
Worked Example
g from a pendulum. A 1.000 ± 0.002 m pendulum swings with a period of 2.007 ± 0.005 s. g = 4π²L/T² = 4π² × 1.000 ÷ 2.007² = 9.8009 m/s². The partial derivatives are ∂g/∂L = 4π²/T² = 9.801 s⁻² and ∂g/∂T = −8π²L/T³ = −9.767 m/s³. The two terms are (9.801 × 0.002)² = 0.000384 and (9.767 × 0.005)² = 0.002385; their sum is 0.002769 and its square root is δg = 0.0526. Rounded: g = 9.80 ± 0.05 m/s². The period gives 86 % of the variance even though its relative error (0.25 %) is close to the length’s (0.20 %), because T is squared.
Density by the product rule. m = 25.30 ± 0.05 g and V = 9.4 ± 0.2 cm³ give ρ = 2.6915 g/cm³. The relative errors are 0.198 % and 2.128 %, so δρ/ρ = √(0.198² + 2.128²) = 2.137 % and δρ = 0.0575 g/cm³: ρ = 2.69 ± 0.06 g/cm³, in line with aluminium at 2.70 g/cm³ (PubChem).
The common mistake: forgetting the power. In the pendulum, T appears as T², so its relative error counts twice: 2 × 0.249 % = 0.498 %. Combined with L’s 0.2 %, that is √(0.2² + 0.498²) = 0.537 %, or ± 0.053 m/s². Counting T’s error once gives √(0.2² + 0.249²) = 0.319 %, or ± 0.031 m/s², which understates the uncertainty by 40 %.
Show Work
Formulas
From Gauss to the GUM
The rule that independent errors combine as the square root of a sum of squares comes from Carl Friedrich Gauss’s theory of observational errors. He used it alongside the method of least squares in Theoria motus (1809), his work on planetary orbits, and set it out in Theoria combinationis observationum erroribus minimis obnoxiae (1821–1823). In German textbooks the formula is still called the Gaussian law of error propagation.
The Monte Carlo method came from Los Alamos after the Second World War. Stanisław Ulam, John von Neumann and Nicholas Metropolis used random sampling to model neutron transport, and Metropolis and Ulam published The Monte Carlo Method in 1949. Instead of linearising a formula, you push thousands of random inputs through it and look at the spread of the outputs.
Both are now standard practice. The Guide to the Expression of Uncertainty in Measurement (the GUM, first published by ISO in 1993 and reissued as JCGM 100:2008) uses the first-order formula on this page, and its Supplement 1 (JCGM 101:2008) uses Monte Carlo for formulas where linearising is not good enough.
About This Tool
This calculator takes any formula you can type (powers, roots, trig, logs, exponentials) and any number of measured inputs with absolute or percentage uncertainties. It finds each partial derivative symbolically with the same engine as the Derivative Calculator. If a derivative has no symbolic form it uses a central-difference numeric one and tells you. It then combines the terms, rounds the uncertainty and the result together, and shows which input contributes most of the error. The Simple rules mode does the same job with the sum, product and power rules taught in first-year labs, and warns you when a variable that appears twice makes those rules wrong. The Monte Carlo mode checks the linear answer against 10,000 random draws.
Inputs are treated as independent, and units are not converted: give every value in consistent units. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Significant Figures Calculator, Scientific Notation Converter, and Dimensional Analysis Checker.
Frequently Asked Questions
What is the formula for error propagation?
For independent measurements, δf = √[(∂f/∂x₁·δx₁)² + (∂f/∂x₂·δx₂)² + …]. For a pendulum, g = 4π²L/T² with L = 1.000 ± 0.002 m and T = 2.007 ± 0.005 s: ∂g/∂L = 9.801 and ∂g/∂T = −9.767, so δg = √[(9.801 × 0.002)² + (9.767 × 0.005)²] = 0.0526, and g = 9.80 ± 0.05 m/s².
How do uncertainties combine when I multiply or divide?
Relative uncertainties add in quadrature. A block of 25.30 ± 0.05 g (0.20 %) and 9.4 ± 0.2 cm³ (2.13 %) has a density of 2.691 g/cm³ with a relative uncertainty of √(0.20² + 2.13²) = 2.14 %, so ρ = 2.69 ± 0.06 g/cm³. The volume causes 99 % of the error.
What happens to the uncertainty when a value is squared?
The relative uncertainty is multiplied by the power. For P = V²/R with V = 12.0 ± 0.1 V (0.83 %) and R = 100 Ω ± 5 %, V² carries 2 × 0.83 = 1.67 %, so P = 1.44 W with √(1.67² + 5²) = 5.27 %, which is 1.44 ± 0.08 W.
How many significant figures should an uncertainty have?
Usually one, or two when the first digit is 1, and the result is then rounded to the same decimal place. 9.80088 ± 0.0526 becomes 9.80 ± 0.05; 102.5 ± 1.495 becomes 102.5 ± 1.5, because rounding 1.495 to one figure (1) would change it by a third.
Why are uncertainties added in quadrature instead of just added?
Independent errors are as likely to cancel as to add, so their squares add, not the errors themselves. A temperature rise of 36.8 ± 0.2 °C minus 21.3 ± 0.2 °C is 15.5 ± 0.28 °C, not ± 0.4 °C; straight addition gives the worst case, which overstates the spread.
How do I use the Error Propagation Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Physics labs
A pendulum gives g = 9.80 ± 0.05 m/s², and the bar chart shows the period causes 86 % of the error, so timing more swings is the way to improve it.
Density and identification
A metal block at 2.69 ± 0.06 g/cm³ is consistent with aluminium (2.70 g/cm³); the volume reading is 99 % of the uncertainty.
Electronics
With 12.0 ± 0.1 V across a 5 % 100 Ω resistor the power is 1.44 ± 0.08 W, and 90 % of that comes from the resistor tolerance.
Sports science
A 0.145 ± 0.001 kg ball at 40.2 ± 0.5 m/s has 117 ± 3 J of kinetic energy; the speed reading causes 93 % of the error.
Measuring areas
A 12.5 ± 0.1 cm by 8.2 ± 0.1 cm card has an area of 102.5 ± 1.5 cm², with 70 % of the uncertainty from the shorter side.
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