How this calculator works
This tool combines independent standard uncertainties in quadrature after scaling each by the function derivative. It is a first-order Taylor approximation near the entered values, not a worst-case bound. Powers include uncertainty in the exponent; logarithm bases, exponential bases and root orders are exact supplied constants. Correlation is assumed zero. Large relative errors, singularities and strongly curved functions can invalidate the approximation. At a stationary point, a zero first-order uncertainty can miss higher-order variation.
Formula
u(f) = sqrt[(df/dx x u(x))² + (df/dy x u(y))²] for independent inputs. Relative uncertainty = u(f)/|f| when f is nonzero.
Worked example
Adding 3 +/- 0.3 and 4 +/- 0.4 gives 7 with standard uncertainty sqrt(0.09 + 0.16) = 0.5. Multiplying 3 +/- 0.1 by 4 +/- 0.2 gives 12 with uncertainty sqrt(0.4² + 0.6²) = 0.721110.
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Frequently asked questions
Are these maximum possible errors?
No. Enter standard uncertainties, interpreted like standard deviations. Worst-case tolerances combine differently.
What if inputs share a calibration error?
They may be correlated. A covariance term is required and is not included by this independent-input tool.
Why reject zero for a square root?
The root derivative is singular at zero. A first-order symmetric uncertainty calculation is unreliable there.