Physics
Error propagation calculator
Type the formula, enter each quantity with its uncertainty, and get the result already rounded the way a lab report writes it, with how much each measurement contributes. Or paste a set of repeated readings for their mean and its error.
Where a result's uncertainty comes from
Every measurement carries an uncertainty, and every quantity computed from measurements inherits it. If you measure a pendulum's length L and period T to find g = 4π²L/T², the uncertainty in g depends on those in L and T, and on how sensitive g is to each. That sensitivity is the partial derivative: ∂g/∂T has a T³ in the denominator, which is why a small error in the period counts twice as much, in percentage terms, as the same error in the length.
There are two rules for combining the contributions. The maximum error adds them in absolute value, Δf = Σ|∂f/∂x|·Δx: the worst case, in which every error pushes the same way, and the one schools start from. Addition in quadrature, σ_f = √Σ(∂f/∂x·σ)², holds for random, independent errors that partly cancel: it is the one universities and laboratories use. It always gives a smaller number, and this page shows both.
For a series of repeated readings of the same quantity, the best value is the mean and its uncertainty is the standard error, σ/√n: the standard deviation says how much a single reading scatters, and divided by √n it says how uncertain the mean of n readings is. With only a few readings the half range, (max − min)/2, is often used instead: rougher, but it takes no arithmetic.
Common mistakes
- Adding absolute uncertainties of quantities that are multiplied or divided: for products and quotients the relative uncertainties combine, which is what the partial derivatives do.
- Forgetting the exponent: in T² the relative uncertainty of T counts twice, in √h it counts half.
- Writing the result with every digit the calculator shows: 9.8107 ± 0.0412 is written 9.81 ± 0.04. Digits beyond the uncertainty carry no information.
- Giving the standard deviation as the uncertainty of the mean: that is the spread of a single reading. The uncertainty of the mean is σ/√n, smaller the more readings you take.
Frequently asked questions
Maximum error or addition in quadrature?
It depends on what the course asks for. The maximum error is cautious and simple, and schools use it. Addition in quadrature is correct for random, independent errors, and it is the standard at university and in the scientific literature. If the errors are not independent, for example because two quantities were measured with the same miscalibrated instrument, neither is enough on its own.
How do I round a result with its uncertainty?
The uncertainty first, to one significant figure, or two if the first figure is a 1 (rounding 0.14 to 0.1 would change the uncertainty by almost a third). Then the value, to the same decimal place as the uncertainty. So 9.8107 ± 0.0412 becomes 9.81 ± 0.04 and 9.8107 ± 0.0147 becomes 9.811 ± 0.015.
What is the contributions table?
It shows how much each quantity weighs on the final uncertainty: the partial derivative, its product with the uncertainty, and its share of the total variance. It tells you where improving the measurement pays: in the pendulum example the period accounts for about 80 %, so timing ten swings instead of one does far more than a more precise ruler.
Why σ/√n and not just σ?
Because they answer different questions. σ is the spread of the readings: how far a single reading typically falls from the mean. σ/√n is the uncertainty of the mean itself, which gets more precise as readings accumulate. With 100 readings the mean is ten times more precise than a single reading.
Can I use constants in the formula?
Yes: π (or pi) and e are recognised, and numbers can be written in scientific notation, such as 6.67e-11. A measured constant with its own uncertainty, such as g, should be entered as a variable instead, so that its uncertainty enters the calculation.
How this calculation works
For a quantity f(x₁, …, xₙ) computed from measurements xᵢ ± σᵢ: the value is f at the measured values; the uncertainty in quadrature is σ_f = √Σ(∂f/∂xᵢ·σᵢ)², for random and independent errors; the maximum error is Δf = Σ|∂f/∂xᵢ|·σᵢ. The partial derivatives are taken symbolically, and numerically by a central difference where no symbolic form is available. Each quantity's share is (∂f/∂xᵢ·σᵢ)²/σ_f². For repeated readings: mean x̄ = Σxᵢ/n, sample standard deviation s = √(Σ(xᵢ − x̄)²/(n − 1)), standard error s/√n, half range (max − min)/2. Rounding: the uncertainty to one significant figure (two if the first is a 1), the value to the same decimal place.