Combining Uncertainties

Results are usually calculated from more than one measurement, so their uncertainties must be combined. The OCR rules are:• p = x + y or p = x − y: add the absolute uncertainties of x and y. (For p = 2x, the absolute uncertainty in x is doubled.)
• p = xy or p = x/y: add the percentage uncertainties of x and y.
• p = xⁿ: multiply the percentage uncertainty of x by n.Uncertainties never cancel: even when subtracting, the absolute uncertainties add.Worked example: Find the density of a metal cube.
Mass m = 50.0 ± 0.5 g, so the percentage uncertainty in m = 0.5 ÷ 50.0 × 100% = 1.0%.
Side x = 2.00 ± 0.02 cm, so the percentage uncertainty in x = 1.0%.
Volume V = x³ = 8.00 cm³, with a percentage uncertainty of 3 × 1.0% = 3.0%.
Density ρ = m ÷ V = 50.0 ÷ 8.00 = 6.25 g cm⁻³.% uncertainty in ρ = 1.0% + 3.0% = 4.0%Absolute uncertainty = 4.0% of 6.25 = 0.25 g cm⁻³, so ρ = 6.25 ± 0.25 g cm⁻³.Subtraction example: x = 10.0 ± 0.2 cm and y = 4.0 ± 0.1 cm give x − y = 6.0 ± 0.3 cm.The rules also show where to improve an experiment: the side length is cubed, so its uncertainty counts three times and measuring it more carefully has the biggest effect.