Result

Uncertainty in a sum or difference

δ z = (δ x)^{2} + (δ y)^{2}

Statement

For z = x ± y, the absolute uncertainties of x and y combine in quadrature, assuming independent, random errors.

Symbols and units

$δ x, δ y$	absolute uncertainties in x, y	same units as x, y
$δ z$	absolute uncertainty in the result	—

When it applies

Independent, randomly distributed errors in x and y.

Don't use when

Does not apply to a product or quotient — use relative uncertainties in quadrature instead (see Uncertainty in a product or quotient).

Derivation

General propagation: each source contributes through its partial derivative, added in quadrature:

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

For z = x ± y, both partial derivatives equal ±1:

δ z = (δ x)^{2} + (δ y)^{2}

Worked examples

Adding two measured lengths

x = 12.0 ± 0.1 cm, y = 5.0 ± 0.2 cm:

z = x + y = 17.0 cm

δ z = 0. 1^{2} + 0. 2^{2} \approx 0.22 cm

Reported as z = 17.0 ± 0.2 cm.

Watch out for

Never simply add absolute uncertainties (δz = δx + δy) — that overstates the error for independent, random sources.

Connects to