Calculate uncertainty propagation through mathematical operations. Supports addition, multiplication, and arbitrary function formulas.
Turn measurements with stated uncertainties into a nominal calculated value and its propagated uncertainty. Select addition or multiplication for common cases, or enter a formula with values and uncertainties for its variables. The tool combines independent uncertainty contributions according to each input's effect on the result. The point most often misread is the combination rule: independent uncertainties are combined in quadrature, not simply added. Thus uncertainties of 0.3 and 0.4 in a sum produce 0.5, not 0.7.
A concrete input and expected output from the current implementation.
Input
Formula: x + y x: value 3, uncertainty 0.3 y: value 4, uncertainty 0.4
Expected output
Result: 7 +/- 0.5
The nominal sum is 3 + 4 = 7. For independent inputs, the propagated uncertainty is sqrt(0.3^2 + 0.4^2) = sqrt(0.25) = 0.5.
For addition z = x + y, nominal values are added while independent absolute uncertainties combine as sqrt(u_x^2 + u_y^2). For multiplication z = x*y, the equivalent first-order form is u_z = |z|*sqrt((u_x/x)^2 + (u_y/y)^2) when both inputs are nonzero. Addition therefore operates on absolute uncertainty, while multiplication depends on fractional uncertainty.
For f(x1, ..., xn), each variable's standard uncertainty is weighted by the local sensitivity of f to that variable. The combined form is u_f = sqrt(sum((partial f / partial xi * u_i)^2)), with sensitivities evaluated at the nominal inputs. This first-order result describes local behavior, not necessarily the full output distribution of a strongly nonlinear formula.
No. For independent standard uncertainties, addition mode squares each uncertainty, adds those squares, and takes the square root. Directly adding the plus-or-minus values describes a different, worst-case bound. If the inputs are correlated, neither interpretation is automatically adequate; covariance or a separate dependence analysis is needed.
Formula mode accepts a supported mathematical expression and propagates the entered variable uncertainties through its local sensitivities. That is suitable for smooth formulas near the supplied values. It is not a reliable substitute for interval or simulation methods when the expression is discontinuous, crosses a domain boundary, or has uncertainty large enough to make the response strongly asymmetric.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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