Adding Percentage Uncertainties

A final result is usually calculated by multiplying or dividing several measurements, for example concentration = moles ÷ volume. Each measurement brings its own uncertainty into the answer. The rule is simple:when measurements are multiplied or divided, add their percentage uncertaintiesThe total percentage uncertainty can then be turned back into an absolute uncertainty in the answer.Worked example: 2.65 g of sodium carbonate (found by difference, ± 0.01 g) is dissolved and made up to 250.0 cm³ in a volumetric flask (± 0.30 cm³). The concentration is 0.100 mol dm^-3. Find its uncertainty.mass: (0.01 ÷ 2.65) × 100 = 0.38%volume: (0.30 ÷ 250.0) × 100 = 0.12%total = 0.38 + 0.12 = 0.50%absolute uncertainty = 0.50% of 0.100 = 0.0005, so the concentration is 0.100 ± 0.0005 mol dm^-3Do not add absolute uncertainties with different units (g and cm³ cannot be added). Converting each to a percentage first makes them comparable.Values that are not measured, such as M_r or the 1000 used to convert cm³ to dm³, are treated as having no uncertainty.Comparing the individual percentages also shows which measurement dominates the total, which tells you where improving the method would have the greatest effect.