Free Physics Tool

Uncertainty
Propagator

Enter your formula and measurements. Get the propagated uncertainty with full step-by-step working - instantly. No signup, no ads, no nonsense.

Built for university physics students, A-level physics students, IB Physics internal assessments, and anyone writing an experimental science lab report. Whether you are working through kinetic energy, Ohm's law, or a custom formula from a physics experiment - enter your formula and measured values and the calculator propagates the measurement uncertainty automatically, combining partial derivatives in quadrature with full step-by-step working shown.

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Tip: Use * for multiply, ^ for powers, ( )^0.5 for square roots

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How uncertainty propagation works

When you measure physical quantities, every measurement carries an uncertainty. When those measurements are combined in a formula, the uncertainties combine too - this process is called uncertainty propagation (also known as error propagation or propagation of errors). It is a fundamental technique in physics lab reports, error analysis, and experimental science at university level.

For a function f(x₁, x₂, ...), the propagated uncertainty is:

Δf = √[ (∂f/∂x₁ · Δx₁)² + (∂f/∂x₂ · Δx₂)² + ... ]

This error propagation calculator computes partial derivatives numerically using the central difference method, so it works for any formula - not just simple products and sums. Whether you need to propagate measurement uncertainty through kinetic energy, Ohm's law, or a custom lab formula, just type it in.

Worked example: kinetic energy

Suppose you are measuring kinetic energy in a physics experiment using KE = 0.5 × m × v². Your laboratory measurements are: mass m = 2.00 ± 0.05 kg, velocity v = 3.00 ± 0.10 m/s.

Step 1 - Compute the result: KE = 0.5 × 2.00 × 3.00² = 9.00 J

Step 2 - Find the partial derivatives: ∂KE/∂m = 0.5 × v² = 4.50 · ∂KE/∂v = m × v = 6.00

Step 3 - Combine in quadrature: ΔKE = √[(4.50 × 0.05)² + (6.00 × 0.10)²] = √[0.0506 + 0.3600] ≈ 0.64 J

Result: KE = 9.00 ± 0.64 J. Velocity dominates the combined standard uncertainty here because it appears squared - small measurement errors in v have a larger effect on the result. This is the standard uncertainty propagation method used in all physics lab reports.

Want to understand the rules for each operation? Read the complete uncertainty propagation guide with worked examples for addition, multiplication, exponents, and more.

Frequently asked questions

Q1What is uncertainty propagation in physics?

Uncertainty propagation - also called error propagation or propagation of errors - is the process of calculating how measurement errors in individual variables combine to produce the total uncertainty in a calculated result. It is essential for physics lab reports at university level.

Q2How do I propagate uncertainty for multiplication?

For a product f = x · y, the relative uncertainty is: Δf/f = √[(Δx/x)² + (Δy/y)²]. This error propagation calculator computes this automatically for any formula.

Q3What is the error propagation formula?

The general formula is Δf = √[ (∂f/∂x₁ · Δx₁)² + (∂f/∂x₂ · Δx₂)² + ... ]. It combines each variable's partial derivative multiplied by its measurement uncertainty, summed in quadrature.

Q4Does this tool work for complex formulas?

Yes. It computes partial derivatives numerically, so it handles any formula including powers, square roots, trigonometric functions, and combinations of variables. Just type it naturally.

Q5How do I calculate uncertainty in a physics lab report?

Identify all measured variables and their uncertainties, write your formula, then apply the uncertainty propagation formula by taking partial derivatives with respect to each variable and combining them in quadrature. This calculator does all of that automatically and shows step-by-step working.

Q6Is this tool free to use?

Completely free, no account required, no limits.

Q7What is the difference between absolute and relative uncertainty?

Absolute uncertainty is expressed in the same units as the measurement - for example ±0.05 m. Relative uncertainty (also called fractional uncertainty) is the ratio of the absolute uncertainty to the measured value - 0.05/2.00 = 2.5%. For addition and subtraction, absolute uncertainties are combined in quadrature. For multiplication and division, relative uncertainties are combined in quadrature.

Q8What is the difference between systematic error and random error?

Random errors are unpredictable fluctuations that scatter measurements around the true value - they are reduced by averaging. Systematic errors are consistent biases that shift every measurement in the same direction - they are not reduced by averaging and must be identified and corrected separately. Standard uncertainty propagation handles random measurement uncertainty only.

Q9How do I propagate uncertainty for addition and subtraction?

For z = x + y or z = x − y, combine absolute uncertainties in quadrature: Δz = √[(Δx)² + (Δy)²]. Both addition and subtraction follow the same rule - the sign of the operation does not affect the uncertainty formula.

Q10How do I propagate uncertainty for exponents and powers?

For z = xⁿ, the relative uncertainty scales with the exponent: Δz/z = |n| × (Δx/x). For example, a 2% uncertainty in radius becomes a 6% uncertainty in a volume formula r³. This is why measurements raised to high powers dominate the combined standard uncertainty.

Q11What does combining uncertainties in quadrature mean?

Combining in quadrature means squaring each uncertainty contribution, summing them, then taking the square root - the same as calculating the hypotenuse of a right triangle. It is the correct method for combining independent random uncertainties because errors in independent variables partially cancel, so the total is always less than the direct sum.

Q12How do I report uncertainty in a physics lab report?

Round the uncertainty to 1 or 2 significant figures first. Then round the result to the same decimal place as the uncertainty. Report as: value ± uncertainty (units). For example: KE = 9.0 ± 0.6 J. Never report more decimal places in the result than in the uncertainty.

Learn how uncertainty propagation works