How to use this calculator
Write the formula with your own variable names, then list each measured input as name = value ± uncertainty. Use standard uncertainties (one standard deviation) for a standard uncertainty in the result. The inputs are assumed independent.
Formulas
Worked example
g = 4π²L/T² with L = 1.000 ± 0.002 m and T = 2.007 ± 0.005 s
- g = 4π²(1.000)/2.007² = 9.801 m/s².
- ∂g/∂L = 4π²/T² = 9.80; ∂g/∂T = −8π²L/T³ = −9.77.
- Contributions: 9.80 × 0.002 = 0.0196 and 9.77 × 0.005 = 0.0488.
- σg = √(0.0196² + 0.0488²) = 0.053, so g = 9.80 ± 0.05 m/s².
Frequently asked questions
Why add in quadrature instead of adding the errors?
Independent random errors rarely all push the same way, so they partly cancel. Adding squares gives the expected spread; the straight sum is a worst-case bound.
When is the linear method not good enough?
When the formula is strongly nonlinear over the uncertainty range, for example a square root near zero or a large relative error. The Monte Carlo check shows this; quote its interval then.
How many digits should I report?
Round the uncertainty to one or two significant figures and the result to the same decimal place, as the calculator does.