Uncertainty Propagation Calculator
Enter your measurements as value ± uncertainty and choose how they combine — a sum, a product, a power, or any formula you type. You get the result with its propagated uncertainty, rounded the way lab reports expect, plus a breakdown of how much each input contributes and the full working.
Type the formula with variables. Use + − * / ^ and brackets, sqrt, sin, cos, tan, asin, acos, atan, ln, log10, exp, abs, pi and e. Write numbers with a decimal point.
Read as: 4 × π^2 × L / T^2
Results
9.80 ± 0.11
Result
1.11%
Relative uncertainty
9.80088 ± 0.109272
Unrounded
Where the uncertainty comes from
| Input | Value ± uncertainty | Sensitivity ∂q/∂x | Contribution |∂q/∂x| × δx | Share of the variance |
|---|---|---|---|---|
| L | 1.000 ± 0.005 | 9.80088 | 0.0490044 | 20.1% |
| T | 2.007 ± 0.010 | -9.76669 | 0.0976669 | 79.9% |
The working
q = 4 × π^2 × L / T^2 = 9.80088
∂q/∂L ≈ 9.80088; |∂q/∂L| × δL = 0.0490044
∂q/∂T ≈ -9.76669; |∂q/∂T| × δT = 0.0976669
δq = √(0.0490044² + 0.0976669²) = 0.109272
Rounded: the uncertainty to one significant figure (two when it starts with 1), and the value to the same decimal place: 9.80 ± 0.11.
Assumes the uncertainties are independent (uncorrelated) and small enough that the formula is close to a straight line across each ± range — first-order propagation, as in introductory lab manuals and the GUM.
How to Use the Uncertainty Propagation Calculator
- 1. Pick the calculation. Sum or difference, product or quotient, and power each apply the textbook rule your lab manual asks you to show. Any formula handles everything else, from a pendulum's g = 4π²L/T² to a projectile's range.
- 2. Enter each measurement as a value and an uncertainty. Use standard uncertainties (± one standard deviation, or the instrument resolution your course tells you to use), all in consistent units. In formula mode, every variable you type gets its own pair of boxes.
- 3. Check how the formula was read. The Read as line writes your formula back with every operator explicit, so 2πr appears as 2 × π × r. Trigonometric functions use radians unless you switch them to degrees.
- 4. Read the result and its breakdown. The headline is rounded to the precision the uncertainty supports; the table shows which measurement contributes most of the uncertainty — the one worth improving.
The Propagation Rules
| Calculation | Rule | Example |
|---|---|---|
| q = a + b or a − b | δq = √(δa² + δb²) | (10.2 ± 0.3) − (4.1 ± 0.4) = 6.1 ± 0.5 |
| q = a × b or a ÷ b | δq/|q| = √((δa/a)² + (δb/b)²) | (5.0 ± 0.1) × (3.0 ± 0.1) = 15.0 ± 0.6 |
| q = xⁿ | δq/|q| = |n| × δx/|x| | (2.0 ± 0.1)³ = 8.0 ± 1.2 |
| q = cx, with c exact | δq = |c| × δx | 2 × (1.50 ± 0.02) = 3.00 ± 0.04 |
| q = f(x, y, …) | δq = √((∂f/∂x · δx)² + (∂f/∂y · δy)² + …) | Any formula |
Independent uncertainties add in quadrature — square, add, square-root — rather than straight. Errors in separate measurements are as likely to pull in opposite directions as in the same one, so they partly cancel; simply adding them gives the worst case, which overstates the uncertainty. The general rule contains every other row of the table: the partial derivative ∂f/∂x, also called the sensitivity coefficient, says how strongly q responds to x, and each input's contribution is that sensitivity times its uncertainty.
Worked Example: g From a Pendulum
A pendulum of length L = 1.000 ± 0.005 m has a period of T = 2.007 ± 0.010 s, and g = 4π²L/T². These are the calculator's default inputs:
- g = 4π² × 1.000 / 2.007² = 9.80088 m/s².
- ∂g/∂L = 4π²/T² ≈ 9.80088, so L contributes 9.80088 × 0.005 = 0.0490044.
- ∂g/∂T = −8π²L/T³ ≈ −9.76669, so T contributes 9.76669 × 0.010 = 0.0976669.
- δg = √(0.0490044² + 0.0976669²) = 0.109272, a relative uncertainty of 1.11%.
- Rounded: g = 9.80 ± 0.11 m/s². The uncertainty starts with a 1, so it keeps two significant figures, and the value is rounded to the same decimal place.
The breakdown is the useful part. The period's relative uncertainty (0.5%) is the same as the length's, but T is squared, which doubles its effect, and T ends up supplying 79.9% of the variance. Timing ten swings and dividing by ten would shrink δT far more cheaply than a better ruler would shrink δL. The analytic result, δg/g = √((δL/L)² + (2δT/T)²), gives the same 0.109.
Why x × x is not the same as x²
The product rule assumes its inputs are independent. Squaring a measurement multiplies it by itself, so both factors carry exactly the same error. Treating (2.0 ± 0.1)² as a product of two independent factors gives a relative uncertainty of √2 × 5% ≈ 7.1%; the power rule correctly gives 2 × 5% = 10%. Enter a repeated measurement once — as a power, or as one variable in a formula.
Assumptions and Limits
- Independent inputs. Correlated measurements — two readings from one miscalibrated instrument, say — need covariance terms that this method leaves out.
- Small uncertainties. The method straightens the formula into a line across each ± range. When the formula curves noticeably over that range, as 1/x does close to zero, the calculator warns you; a Monte Carlo simulation is the honest check then.
- Numerical derivatives. In formula mode each partial derivative is a central difference with a step of about 6 × 10⁻⁶ times the size of the input, accurate to many more digits than any measurement.
- One kind of uncertainty. Combine standard uncertainties with standard uncertainties; do not mix them with maximum ranges or 95% intervals without converting first.
Frequently Asked Questions
- Why are uncertainties added in quadrature?
- Because independent errors are random in size and sign, they partly cancel when combined. Adding their squares and taking the square root gives the standard uncertainty of the result; adding them directly gives an upper bound that is usually far too pessimistic.
- When should I add uncertainties directly instead?
- When your course asks for a maximum (worst-case) uncertainty, or when the errors are fully correlated and always push the same way. Some introductory courses teach the straight sum first; check which your lab manual expects.
- What is relative uncertainty?
- It is the uncertainty divided by the size of the value, δq/|q|, often given as a percentage. Relative uncertainties are what combine in products, quotients and powers, and they let you compare the quality of measurements in different units.
- How many significant figures should an uncertainty have?
- Usually one, or two when the first digit is a 1 (0.11 rather than 0.1), and the value is then rounded to the same decimal place: 9.80 ± 0.11, not 9.80088 ± 0.109272. Some fields always keep two figures; the calculator offers both.
- How does the calculator differentiate my formula?
- It parses the formula itself — no code is executed — evaluates it at your values, and estimates each partial derivative by nudging one variable slightly up and down. For the standard rules (sums, products, powers) it uses the closed-form expressions instead, and the two agree to many decimal places.
- What is the GUM?
- The Guide to the Expression of Uncertainty in Measurement, published by the Joint Committee for Guides in Metrology (which includes the BIPM and ISO), is the reference standard for evaluating and reporting measurement uncertainty. Its law of propagation of uncertainty for independent inputs is the general rule this calculator applies.