Result

Uncertainty in a product or quotient

\frac{δ z}{∣ z ∣} = (\frac{δ x}{x})^{2} + (\frac{δ y}{y})^{2}

Statement

For z = xy or z = x/y, the relative uncertainties of x and y combine in quadrature.

Symbols and units

$δ x / x, δ y / y$	relative uncertainties	—
$δ z /∣ z ∣$	relative uncertainty in the result	—

When it applies

Independent, randomly distributed errors in x and y; z must be non-zero.

Don't use when

Does not apply to a sum or difference — use absolute uncertainties in quadrature instead (see Uncertainty in a sum or difference).

Derivation

Taking logarithms of z = xy turns the product into a sum, and differentiation converts absolute change to relative change:

\frac{δ z}{z} \approx \frac{δ x}{x} + \frac{δ y}{y} (worst case)

For independent, random errors, relative uncertainties add in quadrature rather than linearly:

\frac{δ z}{∣ z ∣} = (\frac{δ x}{x})^{2} + (\frac{δ y}{y})^{2}

Worked examples

Area of a rectangle from measured sides

l = 4.0 ± 0.1 cm, w = 3.0 ± 0.1 cm:

A = l w = 12.0 cm^{2}

\frac{δ A}{A} = (\frac{0.1}{4.0})^{2} + (\frac{0.1}{3.0})^{2} \approx 4.2%

δA ≈ 0.5 cm², so A = 12.0 ± 0.5 cm².

Watch out for

This formula applies to products and quotients only; a sum or difference uses absolute, not relative, uncertainties.

Connects to