Error Propagation Calculator: Uncertainty of Sums, Products, Powers and Functions
Propagate measurement uncertainties through sums, differences, products, quotients, powers, k·Aᵃ·Bᵇ·Cᶜ formulas, logs, exponentials and trig, in quadrature or worst case, with steps.
How it works
General rule: δq = √(Σ (∂q/∂xᵢ · δxᵢ)²) (independent errors) or δq = Σ |∂q/∂xᵢ| · δxᵢ (worst case) · Sum or difference: δq = √(δA² + δB²) · Product or quotient: δq/|q| = √((δA/A)² + (δB/B)²) · Power: δq/|q| = |n| · δA/|A| · k·Aᵃ·Bᵇ·Cᶜ: δq/|q| = √((a·δA/A)² + (b·δB/B)² + (c·δC/C)²) · ln A: δq = δA/|A| · eᴬ: δq = eᴬ·δA · sin A: δq = |cos A|·δA (radians)
Every measured value carries an uncertainty, and any quantity you calculate from it inherits some of it. Error propagation finds that inherited uncertainty from the first-order (linear) change in the result: each input contributes |∂q/∂x| × δx, how strongly the result reacts to that input times that input’s own uncertainty. When the input errors are independent and random, the contributions are added in quadrature (square, add, square root), which is the method in the GUM and most physics and chemistry courses. If you need a guaranteed upper bound, or the errors may all lean the same way, add them directly (worst case), the rule many introductory labs teach first. For sums and differences the absolute uncertainties combine; for products, quotients and powers the relative (percentage) uncertainties combine, multiplied by the size of each exponent. The k · Aᵃ · Bᵇ · Cᶜ mode handles most lab formulas in one go, such as density m/(πr²h) or g = 4π²L/T². The answer is rounded the usual way: uncertainty to one significant figure (two if it starts with 1) and the value to the same decimal place. The uncertainty budget table and chart show which input dominates, so you know what to measure better. Percent error compares a result with a known true value and standard error describes the scatter of a sample mean; this page instead carries known input uncertainties through a formula.
Worked example
Multiply A = 4.52 ± 0.02 by B = 2.0 ± 0.2. The product is q = 4.52 × 2.0 = 9.04. The relative uncertainties are 0.02 ÷ 4.52 = 0.442% and 0.2 ÷ 2.0 = 10%, so in quadrature δq/q = √(0.00442² + 0.1²) = 0.1001 and δq = 9.04 × 0.1001 = 0.9049. Rounded to one significant figure, q = 9.0 ± 0.9 (10% relative). Adding the contributions instead (worst case) gives 0.04 + 0.904 = 0.944.
Sources: LSU Geophysics: Uncertainties and error propagation, part 2 (worked examples, both methods) · Omni Calculator: Error propagation (rod and bird-velocity examples)
Frequently asked questions
How do you propagate uncertainty when multiplying?
Add the relative uncertainties in quadrature: δq/|q| = √((δA/A)² + (δB/B)²), then multiply by |q|. For 4.52 ± 0.02 times 2.0 ± 0.2 that is √(0.0044² + 0.1²) = 10.0%, so 9.04 ± 0.90, reported as 9.0 ± 0.9.
How do you propagate uncertainty when adding or subtracting?
Combine the absolute uncertainties: δq = √(δA² + δB²). For 2.00 ± 0.03 m plus 0.88 ± 0.04 m the sum is 2.88 m and δq = √(0.0009 + 0.0016) = 0.05 m. Subtraction uses the same rule, which is why the difference of two close values can have a huge relative error.
Should I add uncertainties or use quadrature?
Use quadrature when the errors are independent and random; it is the standard method and gives a smaller, more realistic value. Simple addition (worst case) is a safe upper bound and is often taught in introductory labs. The calculator shows both so you can quote what your course expects.
What is the uncertainty of a power or square root?
Multiply the relative uncertainty by the size of the exponent: for q = Aⁿ, δq/|q| = |n| × δA/|A|. A square root halves the relative uncertainty and a square doubles it.
How do I round a result with uncertainty?
Round the uncertainty to one significant figure (two if its first digit is 1), then round the value to the same decimal place. 9.04 ± 0.905 becomes 9.0 ± 0.9, and 0.6667 ± 0.149 becomes 0.67 ± 0.15.
What if my variables are correlated?
The quadrature formula assumes independence. For correlated inputs add the covariance term 2(∂q/∂A)(∂q/∂B)·cov(A, B), or use the worst-case sum as a conservative bound.
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This calculator is for educational purposes. Double-check important results before acting on them.