Propagation of Uncertainties Formulas

Suppose that you have two measurements: [latex]x \pm s_x[/latex] and [latex]y \pm s_y[/latex]

1. Let z be either the sum or the difference of the two measurements:

i.e. [latex]z=x+y[/latex]     or      [latex]z=x-y[/latex]

The standard uncertainty on z is

[latex]s_z=\sqrt{s_x^2+s_y^2}[/latex]

You can extend the argument to more than two variables.

2. Let z be either the product or the quotient of the two measurements:

i.e.  [latex]z=xy[/latex]          or          [latex]z=x/y[/latex]

The fractional standard uncertainty on z is

[latex]\frac{s_z}{z}=\sqrt{\left(\frac{s_x}{x}\right)^2+\left(\frac{s_y}{y}\right)^2}[/latex]

You can extend the argument to more than two variables.

3. Let    [latex]z=cx[/latex]    where c is a constant.

The standard uncertainty on z is

[latex]s_z=cs_x[/latex]

4. There is a general relationship that is useful for all types of functions.

Let z be some function of x        i.e. [latex]z=f(x)[/latex]

The standard uncertainty on z is

[latex]s_z =\frac{df}{dx} s_x[/latex]

Eg.   Power law:  [latex]z=x^n[/latex]

The standard uncertainty on z is    [latex]s_z=n x^{n-1} s_x[/latex]

Eg.   Sine function:    [latex]z = \sin(x)[/latex]

The standard uncertainty on z is      [latex]s_z = (\cos(x)) s_x[/latex]