Error Propagation Calculator

This tool helps you to determine the uncertainty (or error) of any mathematical expression that contains physical quantities with uncertainties. It follows the rules of the Gaussian error propagation: If f is a function of the independent variables X and Y, written as f(X,Y), then the uncertainty in f is obtained by taking the partial derivatives of f with respect to each variable, multiplication with the uncertainty in that variable, and addition of these individual terms in quadrature.

Use "." as decimal mark: 1.234, not 1,234.

How to use

  1. Enter a valid formula using the functions listed at the bottom of this page.
    uncertainty calculator example
  2. In the "quantities with errors" section define all variables which appear in your formula. Use "." as decimal mark, not ",".
    uncertainty calculator example
  3. Click "Evaluate" to obtain the result along with its absolute and relative uncertainty.
    uncertainty calculator example


uncertainty calculator example

with variables a, b, c

Exact error (calculated analytically):

Error calculated by this tool (numerically):

(≅ 0.02 ‰)

Check with Mathematica


This tool is based on numerical methods. However, even for complicated formulas the deviation between numerical and analytical results is usually negligibly small.

Do not use the symbols "^" or "**" for "raised to the power of" and use the "pow(x,y)" function instead. The following math methods are available and can be used in the formula field:

Method Description
acos(x) Returns the arccosine of x, in radians
asin(x) Returns the arcsine of x, in radians
atan(x) Returns the arctangent of x as a numeric value between -PI/2 and PI/2 radians
cos(x) Returns the cosine of x (x is in radians)
exp(x) Returns the value of Ex
log(x) Returns the natural logarithm (base E) of x
pow(x,y) Returns the value of x to the power of y
sin(x) Returns the sine of x (x is in radians)
sqrt(x) Returns the square root of x
tan(x) Returns the tangent of an angle

Euler's constant and Pi are represented by "E" and PI" respectively.


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