Measurements and Uncertainties
This section covers Measurements and Uncertainties. The exam tests SI units and prefixes, scientific notation, significant figures, uncertainty treatment, and graphical analysis, including error bars and gradients.
Core rules
- SI base units include metre, kilogram, second, ampere, kelvin, mole and candela. Derived units are combinations, such as newton = kg m s⁻² and joule = kg m² s⁻².
- For a single reading, an analogue scale is usually quoted to half the smallest division; a digital display is limited by its last digit. State the absolute uncertainty with the same unit and precision as the value.
- For addition or subtraction, add absolute uncertainties. For multiplication or division, add percentage (or fractional) uncertainties. For a power, multiply the percentage uncertainty by the exponent.
- On a graph, plot error bars and find the steepest and shallowest reasonable best-fit lines to estimate gradient uncertainty.
Worked example
A rectangle measures 12.0 ± 0.1 cm by 5.0 ± 0.1 cm. The area is 12.0 × 5.0 = 60 cm². Percentage uncertainties are 0.1/12.0 = 0.83% and 0.1/5.0 = 2.0%, so the area percentage uncertainty is 2.83%. Absolute uncertainty = 60 × 0.0283 = 1.7 cm². Report area = 60 ± 2 cm², or (6.0 ± 0.2) × 10¹ cm².
Traps
Do not add absolute uncertainties when multiplying; convert to percentage first. Do not keep more significant figures than the data justify. Random uncertainty affects precision and can be reduced by repeats; systematic uncertainty shifts all readings and is not reduced by repeats. Always convert prefixes to SI before calculating.