Error Propagation Calculator (uncertainty of sums, products, quotients and powers)

Understanding the error propagation calculator — concept and principle

This calculator finds the uncertainty of a result calculated from two measured values, each with its own standard uncertainty, for sums, differences, products, quotients, powers and multiplication by a constant. Students use it in lab reports, and technicians use it to state how reliable a derived quantity is.

Independent errors add like the sides of a right triangle; B, the less precise input, dominates.

How it is calculated

General law (independent inputs): u_f² = (∂f/∂A)² u_A² + (∂f/∂B)² u_B²

A ± B: u = √(u_A² + u_B²)
A × B or A ÷ B: (u/f)² = (u_A/A)² + (u_B/B)²
Aⁿ: u/f = |n| u_A/A; c × A: u = |c| u_A
Worst case: the contributions are added linearly instead of in quadrature.
Expanded uncertainty U = 2u, giving roughly 95% coverage.

The principle

Each input error pushes the result by an amount equal to the error times the sensitivity of f to that input, its partial derivative. If the two errors are independent, they are as likely to partly cancel as to add, so the combined effect is found like the hypotenuse of a right triangle, by adding squares. Adding them straight is the worst case, which only happens if both errors push the same way. The rules are first-order approximations that work well when the uncertainties are small compared with the values.

Worked example

Default inputs: A = 12.4 ± 0.2, B = 3.1 ± 0.1, operation A × B.

f = 12.4 × 3.1 = 38.44.
Contributions: B × u_A = 3.1 × 0.2 = 0.62 and A × u_B = 12.4 × 0.1 = 1.24.
u_f = √(0.62² + 1.24²) = 1.386, a relative uncertainty of 3.607%.
Worst case = 0.62 + 1.24 = 1.86; expanded (k = 2) = 2.773.
Range at ±1u: 37.0536 … 39.8264, usually reported as 38.4 ± 1.4.

Each point is the calculator re-run with only “Value of A” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Value of A	Result f
3.1	9.61
6.2	19.22
9.3	28.83
12.4	38.44
15.5	48.05
18.6	57.66
24.8	76.88

How to read the result

The relative uncertainty shows which input dominates: here B, with 3.2% relative error, contributes more than A at 1.6%. Improving the worse measurement gives the biggest gain. Round the uncertainty to one or two significant figures and round the result to match.

Common mistakes

Adding absolute uncertainties for a product instead of relative ones.
Using the quadrature rule for errors that are correlated, such as two readings from the same miscalibrated instrument.
Reporting too many digits.
Mixing standard uncertainties with 95% intervals in the same calculation.

Keep in mind: The method assumes independent inputs and small, roughly normal errors. Strongly non-linear functions or large uncertainties need a full analysis, for example a Monte Carlo simulation.

Frequently asked questions

Why add in quadrature?

For independent errors the variances add, and the standard uncertainty is the square root of the variance.

When should I use the worst-case value?

When errors might be correlated or when you need a guaranteed upper bound.

What does k = 2 mean?

A coverage factor of 2 gives an interval that contains the true value with about 95% confidence for normally distributed errors.

Sources: B. N. Taylor and C. E. Kuyatt, NIST Technical Note 1297 (1994), Appendix A and section 6

Checked 9 Nov 2026. NIST Technical Note 1297 (B. N. Taylor & C. E. Kuyatt, 1994), Appendix A “Law of Propagation of Uncertainty”, Eq. (A-3) (first-order Taylor series; covariance terms omitted for independent inputs) and §6 “Expanded Uncertainty” (U = k u_c, k = 2 ≈ 95%). The sum, product and power rules follow from Eq. (A-3).

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